In this question, there are two parts. Let's break them down:

Part (a): Conditions for Inverses

The question asks which function has an inverse. For a function to have an inverse, it must be one-to-one (bijective) rather than two-to-one.

• $g(x) = x^3$ is one-to-one since for every $y$, there is a unique $x$.
• $f(x) = x^2$ is not one-to-one (it fails the horizontal line test because for some $y$, there are two $x$ values).

Thus, the correct choice for a function with an inverse is $g(x) = x^3$. So, this should have been selected.

Part (b): Inverse of $g(x) = x^3$

To find the inverse of $g(x) = x^3$:

1. Start with $y = x^3$.
2. Solve for $x$: $x = \sqrt[3]{y}$.
3. Swap $x$ and $y$ to get the inverse: $y = \sqrt[3]{x}$.

So, the inverse of $g(x) = x^3$ is $g^{-1}(x) = \sqrt[3]{x}$.

If part (b) was asking for the inverse of the chosen function, your input should be $y = x^{-1/3}$ or $y = \sqrt[3]{x}$ (which is the cube root of $x$).

Corrections:

• Part (a): You should choose $g(x) = x^3$, since it has an inverse.
• Part (b): The inverse function should be $y = \sqrt[3]{x}$.

Do you want more details or have any further questions?

Here are 5 related questions:

1. Why can't $f(x) = x^2$ have an inverse over the entire real line?
2. What is the horizontal line test and how does it relate to inverses?
3. Can $f(x) = x^2$ have an inverse if restricted to $x \geq 0$?
4. How do you find the inverse of more complex functions, like $g(x) = x^5 - 2$?
5. What is the geometric meaning of an inverse function?

Tip: When finding the inverse, always check if the function is one-to-one before proceeding with algebraic manipulation.