The function $f(x) = x^2 - 9$ is not one-to-one over its entire domain because it does not pass the Horizontal Line Test; for example, both positive and negative values of $x$ give the same output.

However, if we restrict the domain of $f(x)$, we can make it one-to-one and thereby allow an inverse function $f^{-1}(x)$ to exist. For quadratic functions like $f(x) = x^2 - 9$, restricting the domain to either:

1. $x \geq 0$ (right branch of the parabola) or
2. $x \leq 0$ (left branch of the parabola)

will ensure that the function passes the Horizontal Line Test.

Thus, the domain can be restricted to:

• $x \geq 0$ or
• $x \leq 0$.

Once restricted to either of these domains, the function becomes one-to-one, and the inverse function $f^{-1}(x)$ can be found.

Final Answer:

• To make $f^{-1}(x)$ exist, the domain of $f(x) = x^2 - 9$ should be restricted to either:
  • $x \geq 0$ (non-negative values of $x$), or
  • $x \leq 0$ (non-positive values of $x$).

Would you like details on how the inverse function $f^{-1}(x)$ is calculated? Or do you have any other questions?

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Related Questions:

1. How do you calculate the inverse function for $f(x) = x^2 - 9$?
2. What is the Horizontal Line Test and how does it relate to the existence of an inverse?
3. How does restricting the domain of a quadratic function make it one-to-one?
4. Can all functions have their domain restricted to make an inverse exist?
5. What is the general process of finding the inverse of a quadratic function?

Tip: When restricting a function's domain to make an inverse exist, the choice depends on which part of the graph you want to keep. For quadratic functions, this is often the "positive" or "negative" half of the parabola.