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Let's break down the problem step by step using the graph provided:

### (a) Why is \( f \) one-to-one?
To determine if a function is one-to-one, we check if it passes the **horizontal line test**. A function passes this test if no horizontal line intersects the graph at more than one point. In this case, the graph of \( f \) seems to pass the horizontal line test because any horizontal line would cross the graph only once. 

Thus, the answer is:
**\( f \) is one-to-one because it passes the horizontal line test.**

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### (b) What are the domain and range of \( f^{-1} \)?
The domain and range of the inverse function \( f^{-1} \) are the reverse of the domain and range of \( f \).

- From the graph, the **domain** of \( f \) is the set of all \( x \)-values it covers, which appears to be from approximately \( -2 \) to \( 4 \). So, the range of \( f^{-1} \) will be the same.
- The **range** of \( f \) (the \( y \)-values) appears to be from \( -2 \) to \( 3 \), meaning the domain of \( f^{-1} \) will be from \( -2 \) to \( 3 \).

Therefore, the domain and range of \( f^{-1} \) are:
- Domain: \( [-2, 3] \)
- Range: \( [-2, 4] \)

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### (c) What is the value of \( f^{-1}(0) \)?
To find \( f^{-1}(0) \), we need to find the value of \( x \) such that \( f(x) = 0 \). Looking at the graph, this occurs when \( x = 1 \).

Thus, \( f^{-1}(0) = 1 \).

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### (d) Estimate the value of \( f^{-1}(-1) \) to the nearest tenth.
To estimate \( f^{-1}(-1) \), we need to find the \( x \)-value for which \( f(x) = -1 \). From the graph, it looks like \( f(x) = -1 \) when \( x \approx -0.5 \).

So, \( f^{-1}(-1) \approx -0.5 \).

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### Recap:
- (a) \( f \) is one-to-one because it passes the **horizontal line test**.
- (b) Domain of \( f^{-1} \): \( [-2, 3] \), Range of \( f^{-1} \): \( [-2, 4] \).
- (c) \( f^{-1}(0) = 1 \).
- (d) \( f^{-1}(-1) \approx -0.5 \).

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Do you want more details or have any other questions? Here are 5 related questions to consider:
1. How do you determine if a function is one-to-one using the vertical line test?
2. Can the inverse of a function fail the horizontal line test?
3. What is the relationship between a function and its inverse graphically?
4. How would the function's domain or range change if its graph were shifted up or down?
5. What happens if a function isn't one-to-one? Can it still have an inverse?

**Tip:** When estimating values from a graph, always check nearby points to refine your estimate further.

The graph of function f is given. 
(a) Why is f one-to-one? 
(b) What are the domain and range of f^{-1}? 
(c) What is the value of f^{-1}(0)? 
(d) Estimate the value of f^{-1}(-1) to the nearest tenth.

Explore how to determine if a function is one-to-one using the horizontal line test and learn to calculate the domain and range of the inverse function. This problem also covers finding specific values of the inverse function based on a graph, suitable for Grades 9-12.

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