College Algebra Unit 4 Review: Functions, Logarithms, and Exponentials

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Find the domain of the function "},{"type":"inlineMath","attrs":{"latex":"f(x) = 20 - x"}},{"type":"text","marks":[{"type":"bold"}],"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: Domain of Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to determine the set of all possible input values (domain) for a given function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Domain:"},{"type":"text","text":" The set of all real numbers "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" for which the function "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" is defined."}]}]}]},{"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":"Consider the expression "},{"type":"inlineMath","attrs":{"latex":"20 - x"}},{"type":"text","text":". Is there any value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that would make this expression undefined (such as division by zero or taking the square root of a negative number)?"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Since "},{"type":"inlineMath","attrs":{"latex":"20 - x"}},{"type":"text","text":" is a linear function, think about whether there are any restrictions on "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" for linear functions."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the domain in interval notation, considering your analysis above."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Find the domain of the function "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{1}{x + 4}"}},{"type":"text","marks":[{"type":"bold"}],"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: Domain of Rational Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the domain of a function that involves division, specifically where the denominator cannot be zero."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Rational Function:"},{"type":"text","text":" A function of the form "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{P(x)}{Q(x)}"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"Q(x) \\neq 0"}},{"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":"Set the denominator equal to zero: "},{"type":"inlineMath","attrs":{"latex":"x + 4 = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the value that makes the denominator zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exclude this value from the domain and write the domain 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":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Given "},{"type":"inlineMath","attrs":{"latex":"f(x) = 4x - 3"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"g(x) = 6x - 6"}},{"type":"text","marks":[{"type":"bold"}],"text":", find "},{"type":"inlineMath","attrs":{"latex":"f - g"}},{"type":"text","marks":[{"type":"bold"}],"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: Operations with Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to subtract two functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Function Subtraction:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"(f - g)(x) = f(x) - g(x)"}}]}]}]},{"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":"Write out "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" explicitly."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" from "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"[4x - 3] - [6x - 6]"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Distribute the negative sign and combine like terms."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. 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Use "},{"type":"inlineMath","attrs":{"latex":"y = 3,483,000(2.7)^{0.007t}"}},{"type":"text","marks":[{"type":"bold"}],"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: Exponential Growth"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use an exponential growth model to predict population after a certain number of years."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formula:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Exponential Growth Model:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"y = y_0 a^{kt}"}}]}]}]},{"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":"Determine the value of "},{"type":"inlineMath","attrs":{"latex":"t"}},{"type":"text","text":" (the number of years since 1994)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug "},{"type":"inlineMath","attrs":{"latex":"t"}},{"type":"text","text":" into the formula "},{"type":"inlineMath","attrs":{"latex":"y = 3,483,000(2.7)^{0.007t}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the expression for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. 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