In this problem, the graph of a function $f$ is provided, and you're asked to find the values of the inverse function $f^{-1}$ at specific points.

For inverse functions, recall that if $f(a) = b$, then $f^{-1}(b) = a$. In other words, the inverse function swaps the roles of $x$ and $y$. Thus, to solve for $f^{-1}(2)$, $f^{-1}(4)$, and $f^{-1}(5)$, you need to find where the original function $f(x)$ equals 2, 4, and 5 respectively, and determine the corresponding $x$-values.

Let's go through each part:

1. (a) $f^{-1}(2)$:

   • Look for the $x$-value where $f(x) = 2$. From the graph, when $f(x) = 2$, the corresponding $x$-value is around 2.

2. (b) $f^{-1}(4)$:

   • Find the $x$-value where $f(x) = 4$. According to the graph, $f(x) = 4$ occurs at $x = 0$, so $f^{-1}(4) = 0$.

3. (c) $f^{-1}(5)$:

   • Look for the $x$-value where $f(x) = 5$. The graph doesn't quite reach $y = 5$, so this value does not exist in the range of the graph.

Here are the answers:

• $f^{-1}(2) = 2$
• $f^{-1}(4) = 0$
• $f^{-1}(5)$ does not exist in the graph's range.

Would you like a more detailed explanation?

Related Questions:

1. What are the key characteristics of an inverse function?
2. How can you determine if a function is one-to-one from its graph?
3. What is the graphical interpretation of the inverse function?
4. How does the range of a function affect the existence of its inverse values?
5. How can you verify whether a point lies on the graph of the inverse function?

Tip:

When working with inverse functions graphically, remember that their graphs are reflections over the line $y = x$.