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.