Let's analyze the graph and the expressions presented in the image.

We are given a graph of a function $f$, and we need to evaluate certain expressions related to $f$ and its inverse $f^{-1}$.

Part a. $f(4) = ?$

We are asked to find $f(4)$. This means we need to look at the graph, find $x = 4$, and determine the corresponding $y$-value. From the graph, at $x = 4$, the $y$-value appears to be 4. Therefore, $f(4) = 4$, not 2.

Part b. $f^{-1}(2) = ?$

We need to evaluate $f^{-1}(2)$. This asks for the value of $x$ such that $f(x) = 2$. From the graph, when $y = 2$, the corresponding $x$-value is 2. So, $f^{-1}(2) = 2$, not 0.

Part c. $f^{-1}(-3) = ?$

Here, we are tasked with finding $f^{-1}(-3)$, which asks for the value of $x$ such that $f(x) = -3$. From the graph, when $y = -3$, the corresponding $x$-value is approximately $-3$. Therefore, $f^{-1}(-3) = -3$.

Part d. $f^{-1}(f(391)) = ?$

This expression simplifies using the property that $f^{-1}(f(x)) = x$. Therefore, $f^{-1}(f(391)) = 391$. This part is already correct.

Summary of Corrections:

• a. $f(4) = 4$
• b. $f^{-1}(2) = 2$
• c. $f^{-1}(-3) = -3$
• d. $f^{-1}(f(391)) = 391$ (correct)

Do you want further clarification or any additional details?

Here are 5 related questions to further your understanding:

1. What is the general procedure to find $f(x)$ using a graph?
2. How does the inverse function $f^{-1}(x)$ relate to the original function $f(x)$?
3. Can a function always have an inverse? What conditions must be satisfied?
4. How would you check if a function is one-to-one using its graph?
5. What are the key characteristics of a linear function, such as the one in the graph?

Tip: When dealing with inverse functions, remember that they reverse the effect of the original function, meaning $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.