https://file.aiquickdraw.com/mathbot/file/1726770878212ftpa4tvu.jpg

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.

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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.

Casey is trying to determine whether the two functions f and g are inverses. She composes f with g, generating the graph shown, where f and g are shown as solid blue lines and f(g(x)) is shown as a dashed red line. Which of the following can Casey conclude about f and g?

Explore how to determine if two functions, f and g, are inverses by analyzing their graphs and the composition f(g(x)). This problem introduces key algebraic concepts such as function composition and the inverse function theorem. Suitable for high school students.

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