1. Let's first state the problem: Determine if the function $f(x) = x^2$ has an inverse function. 2. To have an inverse, a function must be one-to-one (bijective), meaning each $y$ value is produced by exactly one $x$ value. 3. Consider the function $f(x) = x^2$. For example, $f(2) = 4$ and $f(-2) = 4$. This means the same $y$ value corresponds to two different $x$ values. 4. Because $f(x)$ is not one-to-one over all real numbers, it does not have an inverse function on its entire domain. 5. However, if we restrict the domain to $x \\geq 0$ or $x \\leq 0$, then the function becomes one-to-one and has an inverse on that restricted domain. Final answer: The function $f(x) = x^2$ does not possess an inverse function over all real numbers, but it does have an inverse if the domain is restricted to $x \\geq 0$ or $x \\leq 0$.