Let's solve the problem step by step. The function provided is $f(x) = x^3 - 5$, which is one-to-one. Here's how we can proceed:

(a) Find the inverse of $f(x)$ and check the answer.

To find the inverse, follow these steps:

1. Replace $f(x)$ with $y$: $y = x^3 - 5$

2. Swap $x$ and $y$ to find the inverse: $x = y^3 - 5$

3. Solve for $y$: $y^3 = x + 5$ $y = \sqrt[3]{x + 5}$

Thus, the inverse function is: $f^{-1}(x) = \sqrt[3]{x + 5}$

Check: To verify, substitute $f(f^{-1}(x))$ and $f^{-1}(f(x))$ to confirm they return $x$:

• $f(f^{-1}(x)) = (\sqrt[3]{x + 5})^3 - 5 = x + 5 - 5 = x$,
• $f^{-1}(f(x)) = \sqrt[3]{(x^3 - 5) + 5} = \sqrt[3]{x^3} = x$.

The inverse is correct.

(b) Find the domain and range of $f$ and $f^{-1}$.

1. Domain of $f(x) = x^3 - 5$:

   • Since $x^3 - 5$ is defined for all real $x$, the domain is: $(-\infty, \infty)$

2. Range of $f(x) = x^3 - 5$:

   • The cubic function $x^3$ spans all real values, and subtracting 5 does not limit the output. Thus, the range is: $(-\infty, \infty)$

3. Domain of $f^{-1}(x) = \sqrt[3]{x + 5}$:

   • The cube root is defined for all real $x + 5$. Thus, the domain is: $(-\infty, \infty)$

4. Range of $f^{-1}(x) = \sqrt[3]{x + 5}$:

   • The cube root also spans all real numbers, so the range is: $(-\infty, \infty)$

(c) Graph $f(x)$, $f^{-1}(x)$, and $y = x$ on the same coordinate axes.

You would graph:

• $f(x) = x^3 - 5$, a cubic function shifted down by 5 units.
• $f^{-1}(x) = \sqrt[3]{x + 5}$, the reflection of $f(x)$ across $y = x$.
• $y = x$, a straight diagonal line passing through the origin.

Answering the multiple-choice questions:

1. Range of $f(x)$: The range of $f(x)$ is all real numbers:

   • Correct answer: $\text{D. The range is the set of all real numbers.}$

2. Domain of $f^{-1}(x)$: The domain of $f^{-1}(x)$ is all real numbers:

   • Correct answer: $\text{D. The domain is the set of all real numbers.}$

Do you want me to graph the functions or explain any specific step further?

Questions to Explore:

1. How do we determine if a function is one-to-one?
2. What is the process of finding the inverse of non-linear functions?
3. Can the domain or range ever be restricted for cubic functions?
4. How does the graph of $f(x)$ relate to its inverse $f^{-1}(x)$?
5. What happens to the graph when constants are added or subtracted to cubic functions?

Tip:

When finding inverses, always check your result by substituting into the original function to ensure they undo each other.