Math 261 Study Guide: Functions, Graphs, and Calculus Concepts

Q2. The entire graph of $f(x)$ is shown in the figure below.

Background

Topic: Function Analysis and Graph Interpretation

This question tests your ability to interpret a function's graph, including domain, range, zeros, intervals of increase/decrease, concavity, function values, and invertibility. These are foundational skills in Calculus for understanding how functions behave.

Key Terms and Concepts:

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

• Range: The set of all possible output values ($f(x)$) the function can take.

• Zeros: Values of $x$ where $f(x) = 0$ (the function crosses the $x$-axis).

• Intervals of Decrease: Where the function's value decreases as $x$ increases.

• Concavity: Whether the graph is 'curved up' (concave up) or 'curved down' (concave down) at a point.

• Invertibility: Whether the function has an inverse (passes the horizontal line test).

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 ($f(x)$) the graph attains.

3. Locate the zeros by finding where the graph crosses the $x$-axis (i.e., where $f(x) = 0$).

4. To find intervals where $f(x)$ is decreasing, observe where the graph moves downward as $x$ increases. Mark these intervals.

5. For concavity at $x = 6$, look at the shape of the graph near this point. Is it 'cup-shaped' (concave up) or 'cap-shaped' (concave down)?

6. To find $f(4)$, locate $x = 4$ on the graph and read the corresponding $y$ value.

7. Consider invertibility: Does any horizontal line intersect the graph more than once? If so, the function is not invertible.

8. Pause here and try to answer each part using the graph before checking the final answers.

Try solving on your own before revealing the answer!

Final Answers:

• A. Domain: $[0, 7]$

• B. Range: $[-2, 5]$

• C. Zeros: $x = 5$

• D. Decreasing Intervals: $(1, 7)$

• E. Concavity at $x = 6$: Concave up

• F. $f(4)$: $1$

• G. Invertibility: Not invertible; fails the horizontal line test.

Each answer is based on careful reading of the graph and understanding of function properties.