1. **State the problem:** Find the inverse function $f^{-1}(x)$ for the function $f(x) = 2x^2 + 4$. 2. **Understand the function:** The function $f(x) = 2x^2 + 4$ is a parabola opening upwards. 3. **Check if the inverse exists:** For a function to have an inverse, it must be one-to-one (pass the horizontal line test). 4. **Horizontal line test:** Since $f(x)$ is a parabola opening upwards, it is not one-to-one over all real numbers because for each $y$ value above 4, there are two $x$ values (positive and negative) that map to it. 5. **Conclusion:** Therefore, the inverse function $f^{-1}(x)$ does not exist over the entire domain of $f$. 6. **Note:** If we restrict the domain to $x \geq 0$ or $x \leq 0$, then the inverse can be found, but without domain restriction, the inverse does not exist.