1. **Problem Statement:** Given the function $f$ with the following properties: - Domain: $[-9, \infty)$ - $x$-intercept(s): $0$ - $y$-intercept: $0$ - Increasing on $(−2, \infty)$ - Decreasing on $(−3, −2)$ - Constant on $(−\infty, −3)$ - Relative minimum at $x = -2$ with value $f(-2) = -9$ - No relative maximum - $f(-8) = -8$ - $f(x) = -9$ at $x = -2$ - $f$ is neither even nor odd 2. **Understanding the behavior:** - The function is constant on $(-\infty, -3)$, so $f(x) = c$ for $x < -3$. - It decreases on $(-3, -2)$, reaching a minimum at $x = -2$. - It increases on $(-2, \infty)$. - Intercepts at zero mean $f(0) = 0$. 3. **Constructing the piecewise function:** - Since $f$ is constant on $(-\infty, -3)$ and $f(-8) = -8$, the constant value is $-8$ there. - On $(-3, -2)$, $f$ decreases from $-8$ to $-9$. - On $(-2, \infty)$, $f$ increases from $-9$ upwards, passing through $(0,0)$. 4. **Example function form:** $$ f(x) = \begin{cases} -8 & x \leq -3 \\ \text{decreasing function} & -3 < x < -2 \\ \text{increasing function} & x \geq -2 \end{cases} $$ 5. **Summary:** The function is piecewise with constant, decreasing, and increasing intervals, a relative minimum at $x=-2$ with value $-9$, and intercepts at zero. **Final answer:** The function $f$ is piecewise defined with the given intervals and properties as described above.