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To make the function \( f(x) = x^2 - 5 \) one-to-one while preserving the same range, we need to restrict its domain so that it behaves in a one-to-one manner. 

### Step-by-Step Solution:

1. **Understanding the function \( f(x) = x^2 - 5 \):**
   - This is a quadratic function, which forms a parabola opening upwards with its vertex at \( (0, -5) \).
   - Without any restriction on the domain, the function is not one-to-one because it’s symmetric around the y-axis; both positive and negative values of \( x \) produce the same output.

2. **Making it one-to-one:**
   - To make \( f(x) \) one-to-one, we can restrict the domain to either \( x \geq 0 \) or \( x \leq 0 \).
   - Restricting to \( x \geq 0 \) will keep the same range \( (-5, \infty) \) while ensuring that each \( y \)-value corresponds to only one \( x \)-value.

3. **Answer:**
   - The domain should be limited to \( x \geq 0 \).

So, to answer the question in the image:

**Set \( x \geq 0 \) to make the function one-to-one and retain the same range.**

Would you like further details on this solution or have additional questions?

### Related Questions:
1. Why does restricting \( x \geq 0 \) make the function one-to-one?
2. How do we determine the range of \( f(x) = x^2 - 5 \)?
3. What would happen if we restricted the domain to \( x \leq 0 \)?
4. Can we apply this domain restriction method to other even functions?
5. How can we check if a function is one-to-one?

**Tip:** A function is one-to-one if it passes the Horizontal Line Test, meaning that any horizontal line drawn on the graph intersects the curve at most once.

How would you limit the domain to make this function one-to-one and still have the same range?

Discover how to restrict the domain of the quadratic function f(x) = x^2 - 5 to make it one-to-one while keeping the same range. This problem explores function inverses, domain restrictions, and the horizontal line test, ideal for students in Grades 10-12.

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