Analyzing Functions and Their Graphs – College Algebra Guidance

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

Background

Topic: Graphs of Functions, Increasing/Decreasing Intervals, Relative Extrema

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

Key Terms and Concepts:

• Domain: All possible $x$-values for which $f(x)$ is defined.

• Range: All possible $y$-values that $f(x)$ attains.

• Zeros: $x$-values where $f(x) = 0$ (where the graph crosses the $x$-axis).

• Intercepts: $x$-intercepts (where $y=0$), $y$-intercept (where $x=0$).

• Increasing/Decreasing Intervals: Where the graph rises or falls as $x$ increases.

• Relative Maximum/Minimum: Highest/lowest points in a local region of the graph.

Step-by-Step Guidance

1. Domain: Look at the leftmost and rightmost $x$-values where the graph exists. Write the interval for $x$.

2. Range: Identify the lowest and highest $y$-values the graph reaches. Express this as an interval for $y$.

3. Zeros of $f$: Find all $x$-values where the graph crosses the $x$-axis (i.e., $y=0$).

4. $f(0)$: Locate the point where $x=0$ and read the corresponding $y$-value (the $y$-intercept).

5. Intervals of Increase/Decrease: Observe where the graph moves upward (increasing) or downward (decreasing) as you move from left to right. Identify the $x$-intervals for each behavior.

6. Values of $x$ for which $f(x) \leq 0$: Find the intervals where the graph is on or below the $x$-axis.

7. Relative Maxima: Identify any peaks (highest points in a local region) and the $x$-values where they occur.

8. Value of $x$ for which $f(x) = 4$: Find where the graph reaches $y=4$ and note the corresponding $x$-value(s).

9. Sign of $f(-1)$: Check the $y$-value at $x=-1$ to determine if it is positive or negative.

Try solving on your own before revealing the answer!

Final Answers:

• a. Domain: $[-5, 5]$

• b. Range: $[-4, 5]$

• c. Zeros of $f$: $x = -4, -1, 3$

• d. $f(0)$: $f(0) = 2$

• e. Increasing on: $(-5, -3)$ and $(1, 3)$

• f. Decreasing on: $(-3, -1)$ and $(3, 5)$

• g. $f(x) \leq 0$ for: $x \in [-5, -4] \cup [3, 5]$

• h. Relative maxima at: $x = -3$ (maximum value $= 5$), $x = 2$ (maximum value $= 3$)

• i. $f(x) = 4$ at: $x = -4$ and $x = -2$

• j. $f(-1)$ is positive.

Each answer is found by carefully reading the graph and applying the definitions above.

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

Background

Topic: Piecewise Functions, Graph Interpretation

This question tests your ability to interpret a piecewise function's graph to determine domain, range, zeros, intercepts, intervals of increase/decrease/constant, and other properties.

Key Terms and Concepts:

• Piecewise Function: A function defined by different expressions over different intervals.

• Domain/Range: As above.

• Zeros: $x$-values where $f(x) = 0$.

• Intercepts: Where the graph crosses the axes.

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

• Sign of $f(x)$: Whether $f(x)$ is positive, negative, or zero for given $x$.

• Even/Odd/Neither: Even: $f(-x) = f(x)$; Odd: $f(-x) = -f(x)$.

Step-by-Step Guidance

1. Domain: Identify the leftmost and rightmost $x$-values where the graph exists.

2. Range: Find the lowest and highest $y$-values the graph attains.

3. Zeros of $f$: Find all $x$-values where the graph crosses the $x$-axis.

4. $f(0)$: Read the $y$-value at $x=0$.

5. Intervals of Increase/Decrease/Constant: Identify where the graph rises, falls, or is flat as $x$ increases.

6. Values of $x$ for which $f(x) > 0$: Find intervals where the graph is above the $x$-axis.

7. Values of $x$ for which $f(x) = -2$: Find $x$-values where the graph is at $y = -2$.

8. Sign of $f(4)$: Check the $y$-value at $x=4$ to see if it is positive or negative.

9. Even, Odd, or Neither: Compare $f(x)$ and $f(-x)$ for symmetry.

10. Is $f(2)$ a relative maximum? Check if $f(2)$ is higher than nearby points.

Try solving on your own before revealing the answer!

Final Answers:

• a. Domain: $[-5, 5]$

• b. Range: $[-2, 2]$

• c. Zeros of $f$: $x = -3, 3$

• d. $f(0)$: $f(0) = 2$

• e. Increasing on: $(-5, -3)$

• f. Decreasing on: $(3, 5)$

• g. Constant on: $(-3, 3)$

• h. $f(x) > 0$ for: $x \in (-3, 3)$

• i. $f(x) = -2$ at: $x = -5$ and $x = 5$

• j. $f(4)$ is negative.

• k. Even function.

• l. $f(2)$ is not a relative maximum.

Answers are based on careful reading of the graph and definitions above.