Let's break down the problem step by step:

Part (a) - Finding the inverse of $f(x) = x^3 + 8$

To find the inverse, we start by expressing $y$ as a function of $x$:

$y = x^3 + 8$

Now, solve for $x$ in terms of $y$:

1. Subtract 8 from both sides: $y - 8 = x^3$

2. Take the cube root of both sides: $x = \sqrt[3]{y - 8}$

Thus, the inverse function $f^{-1}(x)$ is:

$f^{-1}(x) = \sqrt[3]{x - 8}$

Checking the answer

We can verify that the inverse is correct by performing the following checks:

1. Compute $f(f^{-1}(x))$: [ f(f^{-1}(x)) = f(\sqrt[3]{x - 8}) = (\sqrt[3]{x - 8})^3 + 8 = x - 8 + 8 = x ]

2. Compute $f^{-1}(f(x))$: [ f^{-1}(f(x)) = f^{-1}(x^3 + 8) = \sqrt[3]{(x^3 + 8) - 8} = \sqrt[3]{x^3} = x ]

Both compositions yield $x$, confirming that the inverse is correct.

Part (b) - Domain and Range

• The domain of $f(x) = x^3 + 8$ is all real numbers, $(-\infty, \infty)$, because cubic functions are defined for all real values of $x$.

• The range of $f(x) = x^3 + 8$ is also all real numbers, $(-\infty, \infty)$, since cubic functions cover all real values as $x$ increases or decreases without bound.

For the inverse function $f^{-1}(x) = \sqrt[3]{x - 8}$:

• The domain of $f^{-1}(x)$ is $(-\infty, \infty)$, since the cube root function is defined for all real values of $x$.

• The range of $f^{-1}(x)$ is also $(-\infty, \infty)$, as the cube root function can output any real number.

Part (c) - Graphing $f(x)$, $f^{-1}(x)$, and $y = x$

To graph these functions:

1. Graph $f(x) = x^3 + 8$, a cubic function shifted upward by 8 units.
2. Graph $f^{-1}(x) = \sqrt[3]{x - 8}$, which is the reflection of $f(x)$ over the line $y = x$.
3. Plot the line $y = x$ as a reference.

Would you like me to generate the graph for you?

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Do you have any questions or want further details?

Here are 5 related questions:

1. How would the inverse function change if $f(x) = x^3 - 8$?
2. Can you explain why the inverse of a cubic function is also one-to-one?
3. What happens to the domain and range of a function when you reflect it over the line $y = x$?
4. Why do cubic functions always have a domain and range of all real numbers?
5. How would the graph change if we had $f(x) = x^3 + c$ where $c$ is any constant?

Tip: Inverse functions reflect the original function over the line $y = x$, which means the x-values and y-values swap places.