https://file.aiquickdraw.com/mathbot/file/1728419513625mhu0l3u5.jpg

The image shows a graph of a function \( f \), and a question is asked whether \( f \) has an inverse. You correctly indicated "Yes" because the graph passes the horizontal line test (no horizontal line intersects the graph at more than one point), meaning \( f \) is one-to-one and thus has an inverse.

The problem also asks for the values of the inverse function \( f^{-1}(-1) \) and \( f^{-1}(7) \). Here's how to analyze these:

1. **For \( f^{-1}(-1) \)**:
   - \( f^{-1}(-1) = 2 \) is correct. This means that \( f(2) = -1 \), which matches the graph where \( f(2) = -1 \).

2. **For \( f^{-1}(7) \)**:
   - You entered "DNE" (Does Not Exist) for \( f^{-1}(7) \), but this was marked incorrect. To clarify, this means that the function \( f \) does not reach 7 on the \( y \)-axis within the domain shown in the graph. So it seems your understanding was right, but the system expected a different response. Possibly, the expected value might be based on interpreting \( f(3) \) or some error in input handling.

Let me know if you'd like further clarification or if you want to explore the inverse functions in more detail!

### Relative Questions:
1. What does it mean for a function to have an inverse?
2. How can we apply the horizontal line test to determine if a function is invertible?
3. What are the typical notations for expressing inverse functions?
4. How can you determine the inverse of a function from its graph?
5. Why would \( f^{-1}(7) \) be labeled as incorrect in this context?

### Tip:
When solving for the inverse of a function graphically, always ensure that the \( y \)-values match points on the graph for that specific domain.

Does the function f have an inverse? If so, find f^{-1}(-1) and f^{-1}(7).

This problem asks whether a function f has an inverse and requires finding f^{-1}(-1) and f^{-1}(7). Explore key mathematical concepts such as the horizontal line test, graphing functions, and the conditions for the existence of inverse functions. Suitable for Grades 9-12.

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