1. **State the problem:** We have an increasing function $f$ on its domain. We want to know which of the following functions is not necessarily increasing on its domain. 2. **Analyze each option:** - (a) $y = \sqrt{f(x)}$: Since $f$ is increasing and square root is an increasing function for $f(x) \geq 0$, $\sqrt{f(x)}$ is increasing wherever it is defined. - (b) $y = \sqrt[3]{f(x)}$: Cube root is increasing for all real numbers, hence $\sqrt[3]{f(x)}$ is increasing. - (c) $y = [f(x)]^2$: Squaring is not an increasing function for all values. For example, if $f(x)$ is negative and increasing (like from $-2$ to $-1$), then $[f(x)]^2$ goes from $4$ to $1$, which is decreasing. So not necessarily increasing. - (d) $y = e^{f(x)}$: Exponential is an increasing function, so $e^{f(x)}$ is increasing. 3. **Conclusion:** Only option (c) $y = [f(x)]^2$ is not necessarily increasing. **Final answer:** (c)