Study Guide: Analyzing Functions and Their Graphs (College Algebra)

Q51. Analyze the function $f(x)$ shown in the graph:

Background

Topic: Graphs of Functions

This question tests your ability to interpret a function's graph, including domain, range, zeros, function values, intervals of increase/decrease, and relative extrema.

Key Terms and Formulas:

• Domain: The set of all possible $x$ values for which $f(x)$ is defined.

• Range: The set of all possible $y$ values that $f(x)$ can take.

• Zeros: Values of $x$ where $f(x) = 0$ (x-intercepts).

• Intervals of Increase/Decrease: Where the graph rises or falls as $x$ increases.

• Relative Maximum: Highest point in a local region.

• Relative Minimum: Lowest point in a local region.

Step-by-Step Guidance

1. Examine the graph to determine the domain. Look for the leftmost and rightmost $x$ values where the graph exists.

2. Identify the range by finding the lowest and highest $y$ values the graph reaches.

3. Locate the zeros by finding where the graph crosses the $x$-axis.

4. Find $f(0)$ by checking the $y$ value when $x = 0$.

5. Determine intervals where the function is increasing (graph goes up as $x$ increases) and decreasing (graph goes down as $x$ increases).

6. Identify values of $x$ where $f(x) \leq 0$ by observing where the graph is at or below the $x$-axis.

7. Look for any relative maxima and the $x$ values where they occur.

8. Set up how to find the value of $x$ for which $f(x) = 4$ by locating where the graph reaches $y = 4$.

9. Check whether $f(-1)$ is positive or negative by observing the graph at $x = -1$.

Try solving on your own before revealing the answer!

Final Answer:

• a. Domain: $[-5, 5]$

• b. Range: $[-4, 5]$

• c. Zeros: $x = -4, -1, 2, 4$

• d. $f(0)$: $f(0) = 2$

• e. Increasing: $[-5, -3]$, $[1, 3]$

• f. Decreasing: $[-3, -1]$, $[3, 5]$

• g. $f(x) \leq 0$: $x \in [-5, -4] \cup [4, 5]$

• h. Relative maxima: At $x = -3$ ($f(-3) = 5$), $x = 3$ ($f(3) = 4$)

• i. $f(x) = 4$: $x = 3$

• j. $f(-1)$: Negative

Each answer is based on careful reading of the graph. For example, the domain and range are determined by the endpoints and highest/lowest points, zeros are where the graph crosses the $x$-axis, and relative maxima are the highest points in local regions.

Q52. Analyze the function $f(x)$ shown in the graph:

Background

Topic: Graphs of Functions

This question tests your ability to interpret a function's graph, including domain, range, zeros, function values, intervals of increase/decrease/constant, and relative extrema.

Key Terms and Formulas:

• Domain: The set of all possible $x$ values for which $f(x)$ is defined.

• Range: The set of all possible $y$ values that $f(x)$ can take.

• Zeros: Values of $x$ where $f(x) = 0$ (x-intercepts).

• Intervals of Increase/Decrease/Constant: Where the graph rises, falls, or stays flat as $x$ increases.

• Relative Maximum: Highest point in a local region.

• Relative Minimum: Lowest point in a local region.

• Even/Odd/Neither: Even functions are symmetric about the $y$-axis, odd functions are symmetric about the origin.

Step-by-Step Guidance

1. Examine the graph to determine the domain and range by looking at the endpoints and highest/lowest points.

2. Identify the zeros by finding where the graph crosses the $x$-axis.

3. Find $f(0)$ by checking the $y$ value when $x = 0$.

4. Determine intervals where the function is increasing, decreasing, or constant by observing the slope of the graph.

5. Identify values of $x$ where $f(x) > 0$ and where $f(x) = -2$ by checking the graph's position relative to the $x$-axis and specific $y$ values.

6. Check whether $f(4)$ is positive or negative by observing the graph at $x = 4$.

7. Determine if the function is even, odd, or neither by checking for symmetry.

8. Set up how to check if $f(2)$ is a relative maximum by observing the graph at $x = 2$.

Try solving on your own before revealing the answer!

Final Answer:

• a. Domain: $[-5, 5]$

• b. Range: $[-2, 2]$

• c. Zeros: $x = -3, 3$

• d. $f(0)$: $f(0) = 2$

• e. Increasing: $[-5, -3]$

• f. Decreasing: $[3, 5]$

• g. Constant: $[-3, 3]$

• h. $f(x) > 0$: $x \in [-3, 3]$

• i. $f(x) = -2$: $x = -5, 5$

• j. $f(4)$: Negative

• k. Even/Odd/Neither: Even

• l. $f(2)$ a relative maximum? No

The answers are based on careful reading of the graph. For example, the domain and range are determined by endpoints and highest/lowest points, zeros are where the graph crosses the $x$-axis, and symmetry is checked for even/odd status.