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.