1. **State the problem:** We are given the function $f(x) = 14 - x^2$ and asked to find its inverse $f^{-1}(x)$, which can be written in two parts. 2. **Find the inverse function:** To find $f^{-1}(x)$, start by setting $y = 14 - x^2$. 3. Swap $x$ and $y$ to find the inverse: $$x = 14 - y^2$$ 4. Solve for $y$: $$y^2 = 14 - x$$ $$y = \pm \sqrt{14 - x}$$ 5. So the inverse function can be written in two parts: $$f^{-1}(x) = \sqrt{14 - x} \quad \text{and} \quad f^{-1}(x) = -\sqrt{14 - x}$$ 6. **Is the inverse a function?** No, because for some values of $x$, there are two possible outputs (positive and negative square roots), so it fails the vertical line test. 7. **How to modify $f(x)$ so that $f^{-1}(x)$ is a function:** Restrict the domain of $f(x)$ to either $x \geq 0$ or $x \leq 0$. For example, if we restrict $f$ to $x \geq 0$, then the inverse is: $$f^{-1}(x) = \sqrt{14 - x}$$ which is a function. This restriction ensures the inverse passes the vertical line test and is a proper function.