The image you provided depicts a scenario where Casey is analyzing two functions, $f(x)$ and $g(x)$, and their composition, $f(g(x))$, to determine if they are inverse functions. The graph shows:

• The solid blue line represents $f(x)$.
• The solid purple line represents $g(x)$.
• The dashed red line represents the composition $f(g(x))$.

The question asks which conclusion can be drawn about $f$ and $g$, and provides the following options:

A. $f$ and $g$ are inverse functions.

B. The graph of $f$ can be reflected over the line $f(g(x))$ to generate the graph of $g$.

C. The graph of $f(g(x))$ does not indicate that $f$ and $g$ are inverses of each other.

D. The composition $f \circ g$ represents the identity function.

Analysis:

To determine whether $f$ and $g$ are inverse functions, we need to check if their composition $f(g(x))$ equals the identity function $x$. This means that $f(g(x))$ should be a straight line passing through the origin with a slope of 1, i.e., $f(g(x)) = x$.

From the graph:

• The dashed red line, representing $f(g(x))$, does not appear to be a straight line passing through the origin with a slope of 1. Instead, it suggests that $f(g(x))$ is a different function, meaning $f(g(x)) \neq x$.

Thus, $f$ and $g$ are not inverse functions.

Conclusion:

The correct answer is C: The graph of $f(g(x))$ does not indicate that $f$ and $g$ are inverses of each other.

Would you like further clarification or a deeper dive into any part of this analysis?

Here are 5 related questions to expand your understanding:

1. What does the graph of two inverse functions typically look like?
2. How can you verify if two functions are inverses algebraically?
3. Why does the composition $f(g(x))$ need to equal the identity function for $f$ and $g$ to be inverses?
4. What is the geometric relationship between the graphs of inverse functions?
5. How would the graph of $f(x)$ and $g(x)$ change if they were truly inverses?

Tip: Inverse functions reflect over the line $y = x$, and their composition always results in the identity function.