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.