https://file.aiquickdraw.com/mathbot/file/1729452400007k5mnq1ob.jpg

The graph you provided shows the entire function \( f \), and you are asked to graph \( f^{-1} \), the inverse of \( f \).

### How to Graph the Inverse:
To graph the inverse function \( f^{-1}(x) \), you reflect the graph of \( f(x) \) over the line \( y = x \). For each point \( (x, y) \) on the graph of \( f(x) \), the corresponding point on the graph of \( f^{-1}(x) \) will be \( (y, x) \).

#### Steps:
1. **Identify key points on the graph of \( f(x) \):**
   - For example, if \( f(0) = 2 \), then \( f^{-1}(2) = 0 \).
   - Similarly, if \( f(3) = 5 \), then \( f^{-1}(5) = 3 \).

2. **Plot the points for the inverse:**
   - Take several points from the graph and swap the coordinates.

3. **Draw the reflected graph:**
   - After plotting the points, connect them in the same pattern but now reflected across the line \( y = x \).

Would you like me to provide a step-by-step breakdown of some points and their reflections for this graph? Feel free to ask for further details.

Here are 5 questions to expand on the concept:

1. How can you confirm if two functions are inverses of each other?
2. What properties must a function have for its inverse to be a function as well?
3. What is the graphical significance of the line \( y = x \) in finding inverses?
4. Can all functions have inverses? Why or why not?
5. How can you algebraically determine the inverse of a function if given the formula of \( f(x) \)?

**Tip:** Remember, a function must be **one-to-one** (each \( y \)-value has only one corresponding \( x \)-value) for it to have an inverse function.

Below is the entire graph of function f. Graph f^{-1}, the inverse of f.

Learn how to graph the inverse of a function by reflecting it over the line y = x. This problem demonstrates the graph of a function and how to graph its inverse by switching coordinates for key points. Suitable for high school students studying algebra and functions.

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