https://file.aiquickdraw.com/mathbot/file/1728707349611x5l4rexg.jpg

1. 𝑓−1(−4)=
 
2. 𝑓−1(−3)=
 

3. 𝑓−1(0)=
 

4. 𝑓−1(2)=
 

5. 𝑓−1(4)=

Let's analyze the image, which appears to be a graph of a function \( f(x) \). To answer the questions regarding \( f^{-1}(y) \), or the inverse function, we need to find the \( x \)-values for specific \( y \)-values (since \( f^{-1}(y) = x \) if and only if \( f(x) = y \)).

Here’s the breakdown based on the graph:

1. \( f^{-1}(-4) \): This is the \( x \)-value where \( y = -4 \). From the graph, it appears that the point where \( y = -4 \) corresponds to \( x = -1 \). So, \( f^{-1}(-4) = -1 \).

2. \( f^{-1}(-3) \): The \( x \)-value where \( y = -3 \) corresponds to \( x = 0 \) from the graph. So, \( f^{-1}(-3) = 0 \).

3. \( f^{-1}(0) \): The point where \( y = 0 \) corresponds to \( x = 2 \). Thus, \( f^{-1}(0) = 2 \).

4. \( f^{-1}(2) \): For \( y = 2 \), there doesn’t seem to be a corresponding point on the visible part of the graph, so it's likely undefined within this range.

5. \( f^{-1}(4) \): For \( y = 4 \), similar to \( y = 2 \), there’s no corresponding \( x \)-value on the visible graph.

To summarize:
1. \( f^{-1}(-4) = -1 \)
2. \( f^{-1}(-3) = 0 \)
3. \( f^{-1}(0) = 2 \)
4. \( f^{-1}(2) \) is undefined
5. \( f^{-1}(4) \) is undefined

Would you like further details or clarification?

Here are five questions related to this graph and inverse functions:
1. How is the domain of \( f^{-1}(x) \) determined from the original function?
2. What does it mean for the function if \( f^{-1}(x) \) is undefined for certain values?
3. How would the function’s behavior change if the graph extended beyond the current range?
4. What properties must the function \( f(x) \) have to ensure that \( f^{-1}(x) \) exists?
5. How would you calculate the inverse function \( f^{-1}(x) \) algebraically if given a formula for \( f(x) \)?

**Tip**: When looking at graphs of functions and their inverses, reflect the graph over the line \( y = x \) to visualize the inverse relationship more easily!

1. f^{-1}(-4)=
2. f^{-1}(-3)=
3. f^{-1}(0)=
4. f^{-1}(2)=
5. f^{-1}(4)=

Learn how to determine the inverse function values f^{-1}(y) for given y-values by interpreting a function graph. This problem focuses on inverse functions and graph analysis, suitable for students in Grades 9-12.

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