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.

(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: $[-2, 4]$

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

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

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

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.