https://file.aiquickdraw.com/mathbot/file/17274661550640j65a2l6.jpg

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 \( f^{-1}(f(x)) = x \).

Evaluate the function f and its inverse f^{-1} using the graph shown in the image.

Analyze a linear function and its inverse using a graph. Solve expressions such as f(4) and f^{-1}(2). Suitable for students in Grades 10-12.

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