College Algebra Study Guide: Functions, Graphs, and Difference Quotients

{"type":"doc","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"{\"type\":\"doc\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q1. Use the graph to determine:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Open intervals on which the function is increasing, if any.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Open intervals on which the function is decreasing, if any.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) Open intervals on which the function is constant, if any.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Increasing, Decreasing, and Constant Intervals of Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to interpret a function's graph and identify where the function is increasing, decreasing, or constant using interval notation.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Increasing Interval:\"},{\"type\":\"text\",\"text\":\" Where the graph rises as you move left to right.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Decreasing Interval:\"},{\"type\":\"text\",\"text\":\" Where the graph falls as you move left to right.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Constant Interval:\"},{\"type\":\"text\",\"text\":\" Where the graph is flat (horizontal) as you move left to right.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Interval Notation:\"},{\"type\":\"text\",\"text\":\" Use parentheses for open intervals, e.g., \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(a, b)\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Examine the graph and identify the sections where the function moves upward, downward, or remains flat as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" increases.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each section, note the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values that define the start and end of the interval.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the intervals in proper interval notation, using parentheses to indicate open intervals.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check if there are any sections where the function is constant (horizontal line segments).\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) The function is increasing on the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(0, \\\\infty)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) The function is decreasing on the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, 0)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) The function is never constant (no constant interval).\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"We determined these intervals by observing the direction of the graph as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" increases.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q2. Use the graph to determine:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Open intervals on which the function is increasing, if any.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Open intervals on which the function is decreasing, if any.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) Open intervals on which the function is constant, if any.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Increasing, Decreasing, and Constant Intervals of Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question is similar to Q1 and tests your ability to read a graph and identify intervals of increase, decrease, or constancy.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"See Q1 for definitions.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Look at the graph and find where the function rises as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" increases. Note the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-values for these intervals.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify any sections where the function falls as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" increases.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check for any flat (constant) sections.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Express your answers in interval notation.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) The function is increasing on the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(1, \\\\infty)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) The function is never decreasing.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) The function is never constant.\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"These intervals are determined by analyzing the slope of the graph in each region.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q3. Use the graph to determine:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Open intervals on which the function is increasing, if any.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Open intervals on which the function is decreasing, if any.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) Open intervals on which the function is constant, if any.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Increasing, Decreasing, and Constant Intervals of Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question continues to test your understanding of how to read a function's graph for intervals of increase, decrease, or constancy.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"See Q1 for definitions.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify any sections where the graph rises (increasing), falls (decreasing), or is flat (constant).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Mark the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values that define the boundaries of these intervals.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write your answers in interval notation.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) The function is never increasing.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) The function is decreasing on the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-2, \\\\infty)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) The function is constant on the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, -2)\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"These intervals are found by observing the direction of the graph in each region.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q4. The graph of a function \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" is given. Use the graph to find:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) The numbers, if any, at which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" has a relative maximum. What are these relative maxima?\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) The numbers, if any, at which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" has a relative minimum. What are these relative minima?\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Relative Maxima and Minima\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to identify local (relative) maximum and minimum points on a function's graph.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Relative Maximum:\"},{\"type\":\"text\",\"text\":\" A point where the function changes from increasing to decreasing; the highest point in a local region.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Relative Minimum:\"},{\"type\":\"text\",\"text\":\" A point where the function changes from decreasing to increasing; the lowest point in a local region.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Look for peaks (relative maxima) and valleys (relative minima) on the graph.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Record the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values where these occur.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find the corresponding \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-values (function values) at these \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"List all relative maxima and minima as ordered pairs or as separate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\" values.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Relative maximum at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 3\"}},{\"type\":\"text\",\"text\":\" with value $5"},{"type":"inlineMath","attrs":{"latex":".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Relative minima at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 2\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 4\"}},{\"type\":\"text\",\"text\":\" with values $1\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" and $1\"}},{\"type\":\"text\",\"text\":\" respectively.\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"These points are found by identifying where the graph changes direction.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q5. The graph and equation of the function \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" are given: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 2x^3 + 6x^2 - 18x + 8\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Use the graph to find any values at which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" has a relative maximum, and use the equation to calculate the relative maximum for each value.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Use the graph to find any values at which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" has a relative minimum, and use the equation to calculate the relative minimum for each value.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Relative Extrema of Polynomial Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to find relative maxima and minima using both the graph and the function's equation.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Relative Maximum/Minimum:\"},{\"type\":\"text\",\"text\":\" See Q4.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Critical Points:\"},{\"type\":\"text\",\"text\":\" Points where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f'(x) = 0\"}},{\"type\":\"text\",\"text\":\" or \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f'(x)\"}},{\"type\":\"text\",\"text\":\" is undefined.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"First Derivative:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f'(x)\"}},{\"type\":\"text\",\"text\":\" gives the slope of the function; set \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f'(x) = 0\"}},{\"type\":\"text\",\"text\":\" to find critical points.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find the first derivative of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\": \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f'(x) = 6x^2 + 12x - 18\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f'(x) = 0\"}},{\"type\":\"text\",\"text\":\" and solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" to find critical points.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Use the second derivative or the graph to determine if each critical point is a maximum or minimum.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values of the maxima and minima into \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\" to find the corresponding \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-values.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Relative maximum at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = -3\"}},{\"type\":\"text\",\"text\":\" with value $62"}},{"type":"text","text":".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Relative minimum at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 1\"}},{\"type\":\"text\",\"text\":\" with value \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-2\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"We used calculus to find the critical points and evaluated the function at those points.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q6. Using the given graph of the function \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", determine whether the function is even, odd, or neither.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Symmetry of Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to determine if a function is even, odd, or neither by analyzing its graph.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Even Function:\"},{\"type\":\"text\",\"text\":\" Symmetric with respect to the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-axis; \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(-x) = f(x)\"}},{\"type\":\"text\",\"text\":\" for all \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" in the domain.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Odd Function:\"},{\"type\":\"text\",\"text\":\" Symmetric with respect to the origin; \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(-x) = -f(x)\"}},{\"type\":\"text\",\"text\":\" for all \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" in the domain.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Neither:\"},{\"type\":\"text\",\"text\":\" If the function does not have either symmetry.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Examine the graph for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-axis symmetry (even) or origin symmetry (odd).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check if for every point \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x, y)\"}},{\"type\":\"text\",\"text\":\", there is a corresponding \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-x, y)\"}},{\"type\":\"text\",\"text\":\" (even) or \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-x, -y)\"}},{\"type\":\"text\",\"text\":\" (odd).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"If neither symmetry is present, the function is neither even nor odd.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The function is odd.\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The graph is symmetric with respect to the origin.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q7. Use possible symmetry of the graph to determine whether it is the graph of an even function, an odd function, or a function that is neither even nor odd.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Symmetry of Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question is similar to Q6 and tests your ability to recognize function symmetry from a graph.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"See Q6 for definitions.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Look for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-axis symmetry (even) or origin symmetry (odd) in the graph.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check if the function matches the criteria for even, odd, or neither.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The function is even.\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The graph is symmetric with respect to the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-axis.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q8. Use the graph to find the following:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) The domain of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) The range of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) The \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-intercepts\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(d) The \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-intercept\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(e) Intervals on which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" is increasing\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(f) Intervals on which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" is decreasing\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(g) Intervals on which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" is constant\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(h) The number at which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" has a relative minimum\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(i) The relative minimum of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(j) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(-1)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(k) The values of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" for which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = -3\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(l) Is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" even, odd, or neither?\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Graph Analysis and Function Properties\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to extract key information from a function's graph, including intercepts, intervals, and symmetry.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Domain:\"},{\"type\":\"text\",\"text\":\" All possible \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values for which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\" is defined.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Range:\"},{\"type\":\"text\",\"text\":\" All possible \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-values that \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\" can take.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"-intercept:\"},{\"type\":\"text\",\"text\":\" Where the graph crosses the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-axis (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 0\"}},{\"type\":\"text\",\"text\":\").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"-intercept:\"},{\"type\":\"text\",\"text\":\" Where the graph crosses the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-axis (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 0\"}},{\"type\":\"text\",\"text\":\").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Increasing/Decreasing/Constant Intervals:\"},{\"type\":\"text\",\"text\":\" See Q1.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Relative Minimum:\"},{\"type\":\"text\",\"text\":\" See Q4.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Even/Odd/Neither:\"},{\"type\":\"text\",\"text\":\" See Q6.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Read the graph to determine the domain and range, using interval notation.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify where the graph crosses the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-axis and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-axis for intercepts.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Observe the graph to find intervals of increase, decrease, and constancy.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Locate the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-value where the function reaches its lowest point (relative minimum) and the corresponding \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-value.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(-1)\"}},{\"type\":\"text\",\"text\":\" by finding the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-value at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = -1\"}},{\"type\":\"text\",\"text\":\" on the graph.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find all \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = -3\"}},{\"type\":\"text\",\"text\":\" by locating the points where the graph crosses \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = -3\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check the graph for symmetry to determine if the function is even, odd, or neither.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Domain: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, \\\\infty)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Range: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"[-4, \\\\infty)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-intercepts: "},{"type":"inlineMath","attrs":{"latex":"3\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" and $7\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(d) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-intercept: $21"}},{"type":"text","text":"\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(e) Increasing on \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(5, \\\\infty)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(f) Decreasing on \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(0, 5)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(g) Constant on \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, 0)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(h) Relative minimum at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 5\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(i) Relative minimum value: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-4\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(j) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(-1) = 21\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(k) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = -3\"}},{\"type\":\"text\",\"text\":\" at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 4\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 6\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(l) The function is neither even nor odd.\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"All answers are based on careful reading of the graph and applying the definitions.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q9. Use the graph of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" to determine each of the following:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) The domain of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) The range of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) The zeros of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(d) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(5.5)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(e) The intervals on which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" is increasing\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(f) The intervals on which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f\"}},{\"type\":\"text\",\"text\":\" is decreasing\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(g) The values for which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) \\\\leq 0\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(h) Any relative maxima or minima\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(i) The value(s) of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" for which \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 3\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(j) Is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(4)\"}},{\"type\":\"text\",\"text\":\" positive or negative?\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Graph Analysis and Function Properties\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question is similar to Q8 and tests your ability to extract information from a function's graph.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"See Q8 for definitions.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Use the graph to determine the domain and range in interval notation.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values where the graph crosses the $x$-axis (zeros).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Read the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-value at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 5.5\"}},{\"type\":\"text\",\"text\":\" for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(5.5)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify intervals where the function is increasing or decreasing.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find all \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) \\\\leq 0\"}},{\"type\":\"text\",\"text\":\" by looking below or on the $x$-axis.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Locate any relative maxima or minima by finding peaks or valleys.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find all \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 3\"}},{\"type\":\"text\",\"text\":\" by checking where the graph crosses \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = 3\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check the sign of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(4)\"}},{\"type\":\"text\",\"text\":\" by reading the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-value at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 4\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) Domain: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, 6]\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) Range: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, 3]\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) Zeros: "},{"type":"inlineMath","attrs":{"latex":"0\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" and $6\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(d) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(5.5) = 1\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(e) Increasing on \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, 3)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(f) Decreasing on \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(3, 6)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(g) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) \\\\leq 0\"}},{\"type\":\"text\",\"text\":\" for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" in \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, 0) \\\\cup (6, 6]\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(h) Relative maximum at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 3\"}},{\"type\":\"text\",\"text\":\" with value $3"}},{"type":"text","text":"; no relative minimum.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(i) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 3\"}},{\"type\":\"text\",\"text\":\" at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 3\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(j) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(4)\"}},{\"type\":\"text\",\"text\":\" is negative.\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"All answers are based on interpreting the graph and applying function properties.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q10. Evaluate the piecewise function at the given values:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = \\\\begin{cases} 3x + 5 & \\\\text{if } x < 0 \\\\\\\\ 2x + 7 & \\\\text{if } x \\\\geq 0 \\\\end{cases}\"}}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(-1)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(0)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(3)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Piecewise Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate a piecewise-defined function at specific values of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Piecewise Function:\"},{\"type\":\"text\",\"text\":\" A function defined by different expressions for different intervals of the domain.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each value of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\", determine which piece of the function applies (check the condition on $x$).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug the value of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" into the appropriate expression.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify to find the function value.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(a) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(-1) = 2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(b) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(0) = 7\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"(c) \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(3) = 13\"}}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Each value is found by substituting into the correct piece of the function.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q11. The domain of the piecewise function is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, \\\\infty)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\". Graph the function and use your graph to determine the function's range:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = \\\\begin{cases} x + 2 & \\\\text{if } x < 4 \\\\\\\\ x - 2 & \\\\text{if } x \\\\geq 4 \\\\end{cases}\"}}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Piecewise Functions and Range\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to graph a piecewise function and determine its range.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Concepts:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Range:\"},{\"type\":\"text\",\"text\":\" All possible output values (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-values) of the function.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Graph each piece of the function on its respective interval.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Observe the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-values covered by the graph as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" ranges over \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, \\\\infty)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Write the range in interval notation.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The range of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\" is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(-\\\\infty, \\\\infty)\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Both pieces are linear and together cover all real \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y\"}},{\"type\":\"text\",\"text\":\"-values.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q12. Find and simplify the difference quotient for the given function:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = 6\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{f(x + h) - f(x)}{h}\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"h \\\\neq 0\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Difference Quotient\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to compute and simplify the difference quotient, which is foundational for understanding derivatives.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Difference Quotient:\"},{\"type\":\"text\",\"text\":\" \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\dfrac{f(x + h) - f(x)}{h}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\