Q2. The entire graph of 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 concepts in Calculus, especially for understanding how functions behave and how to analyze their properties visually.

Key Terms and Formulas:

• Domain: The set of all -values for which is defined.

• Range: The set of all possible -values (outputs) of .

• Zeros: Values of where (where the graph crosses the -axis).

• Intervals of Decrease: Intervals where the graph is moving downward as increases.

• Concavity: Whether the graph is "curving up" (concave up) or "curving down" (concave down) at a given point.

• Invertibility: Whether the function has an inverse, often checked by the horizontal line test.

Step-by-Step Guidance

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

2. Identify the range by finding the lowest and highest -values the graph attains.

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

4. Find intervals where is decreasing by observing where the graph slopes downward as increases.

5. To determine concavity at , look at the shape of the graph near that point: is it curving upwards or downwards?

6. Find by locating on the graph and reading the corresponding -value.

7. Assess invertibility by considering whether any horizontal line would intersect the graph more than once.

8. For each part, write down your observations and reasoning based on the graph, but stop before stating the final answers.

Try solving on your own before revealing the answer!