https://file.aiquickdraw.com/mathbot/file/1727445836938akwj8zkd.jpg

The graph in the image shows a function \( y = f(x) \), represented by the solid blue line with blue points, and you are asked to sketch the graph of its inverse, \( y = f^{-1}(x) \), and explain why \( f^{-1} \) is also a function.

### Inverse of a Function and Symmetry:
The graph of the inverse function, \( y = f^{-1}(x) \), is a reflection of the graph of \( y = f(x) \) across the line \( y = x \) (shown by the yellow dashed line in the graph). This means that if a point \( (a, b) \) is on the graph of \( y = f(x) \), then the point \( (b, a) \) will be on the graph of \( y = f^{-1}(x) \).

To sketch the inverse function:
1. Reflect each of the blue points on the graph of \( f(x) \) across the line \( y = x \). For instance, if a point on \( f(x) \) is \( (2, 3) \), its corresponding point on \( f^{-1}(x) \) will be \( (3, 2) \).
2. Connect these reflected points, creating the graph of \( f^{-1}(x) \). It should form a curve that is symmetric to the original graph with respect to the line \( y = x \).

### Why \( f^{-1}(x) \) is a function:
For \( f^{-1}(x) \) to be a function, each input \( x \) should have exactly one output \( y \). In the case of the graph shown:
- The original function \( f(x) \) passes the horizontal line test, meaning that any horizontal line will intersect the graph of \( f(x) \) at most once. This ensures that \( f^{-1}(x) \) is also a function, as each input \( y \) (from the original function) maps to exactly one output \( x \) in the inverse.

Would you like more details or have any questions? Here are some related questions:

1. What is the process of finding the inverse of a linear function algebraically?
2. How does the horizontal line test help determine if a function has an inverse that is also a function?
3. Can the inverse of every function be a function? Why or why not?
4. How do reflections across the line \( y = x \) apply to other types of functions, like quadratics?
5. What happens to the domain and range of \( f(x) \) when considering \( f^{-1}(x) \)?

**Tip:** To check if a function is invertible, use the horizontal line test—if no horizontal line intersects the function's graph more than once, the function has an inverse.

The graph of y = f(x) is shown below. Sketch the graph of its inverse, y = f^{-1}(x). Explain why f^{-1} is a function based on its graph.

Learn how to sketch the inverse of a function by reflecting it across the line y = x. This problem demonstrates the properties of inverse functions and explains why the inverse can also be a function based on the graph's symmetry and the horizontal line test. Suitable for students in Grades 9-12.

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