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The question asks which of the given functions has an inverse that is also a function. To determine this, we need to check whether the function is **one-to-one** (injective), which is a necessary condition for a function to have an inverse that is also a function.

Let's go through each option:

### Option A: \( f(x) = 5x^2 - 10 \), with \( x \leq 5 \)
- This is a quadratic function, and quadratic functions are generally not one-to-one because they are symmetric. However, restricting the domain to \( x \leq 5 \) includes only the left side of the parabola (which is decreasing). Therefore, in this domain, the function is one-to-one and would have an inverse that is also a function.

### Option B: \( f(x) = 2x^2 + 1 \), with \( x \geq 1 \)
- Again, this is a quadratic function. Restricting the domain to \( x \geq 1 \) selects only the right side of the parabola (which is increasing). Since the function is strictly increasing in this domain, it is one-to-one and would have an inverse that is also a function.

### Option C: \( f(x) = 2x^2 \), with \( x \geq -2 \)
- This is a quadratic function, and restricting the domain to \( x \geq -2 \) includes both increasing and decreasing parts (the entire parabola). Therefore, the function is not one-to-one over this domain, and it does not have an inverse that is a function.

### Option D: \( f(x) = x^2 + 3 \), with \( x \leq 3 \)
- This is a quadratic function, and restricting the domain to \( x \leq 3 \) selects only the left side of the parabola (which is decreasing). As the function is one-to-one on this domain, it would have an inverse that is also a function.

### Conclusion:
The correct answers are **A**, **B**, and **D** because each of these functions is one-to-one on the given domain, and therefore has an inverse that is also a function.

Would you like more details or have any questions?

Here are 5 follow-up questions related to this concept:
1. What does it mean for a function to be one-to-one (injective)?
2. How can you determine if a function is one-to-one by looking at its graph?
3. What are the steps to find the inverse of a function?
4. Why does restricting the domain of a quadratic function allow it to have an inverse?
5. Can every function have an inverse, and why or why not?

**Tip:** To check if a function has an inverse that is also a function, the **horizontal line test** can be used. If no horizontal line intersects the graph more than once, the function is one-to-one.

Which of the following functions has an inverse that is also a function?

Learn how to determine which quadratic functions have inverses that are also functions. This problem involves understanding quadratic functions, domain restrictions, and applying the horizontal line test. Suitable for students in Grades 9-12.

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