2. Given the following information about a function: - $\mathrm{D}=\{x \in \mathbf{R}\}$ - $\mathrm{R}=\{y \in \mathbf{R} \mid y \geq-2\}$ - decreasing on the interval $(-\infty, 0)$ - increasing on the interval $(0, \infty)$ a) What is a possible parent function? b) Draw a possible graph of the function. c) Describe the transformation that was performed.

Step 1 ：Given the domain and range of the function, as well as its behavior on the intervals $(-\infty, 0)$ and $(0, \infty)$, we can infer a possible parent function.

Step 2 ：The function $f(x) = x^3$ is a cubic function that is decreasing on the interval $(-\infty, 0)$ and increasing on the interval $(0, \infty)$, which matches the given conditions.

Step 3 ：However, the range of $f(x) = x^3$ is all real numbers, which does not match the given range $\{y \in \mathbf{R} \mid y \geq -2\}$.

Step 4 ：To adjust the range, we can shift the function upward by 2 units. This gives us the function $f(x) = x^3 + 2$, which has the desired range.

Step 5 ：So, a possible parent function is $f(x) = x^3 + 2$.

Step 6 ：To draw a possible graph of the function, we plot the function $f(x) = x^3 + 2$. The graph is a cubic curve that passes through the points $(0, 2)$, $(1, 3)$, and $(-1, 1)$, and it is decreasing for $x < 0$ and increasing for $x > 0$.

Step 7 ：The transformation that was performed on the parent function $f(x) = x^3$ to obtain the function $f(x) = x^3 + 2$ is a vertical shift upward by 2 units.

Step 8 ：\(\boxed{f(x) = x^3 + 2}\) is the final answer.