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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 \( y = \sqrt[3]{x} \).

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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.

For a function to have an inverse, it must be one-to-one. Which one of the following functions has an inverse?

Explore which functions have inverses by understanding the one-to-one condition and applying the horizontal line test. This problem illustrates the concepts with the functions x^3 and x^2, detailing why x^3 has an inverse and x^2 does not.

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