Question

Use the graph below of a polynomial function $y=f\left(x\right)$to solve

$$\begin{gathered} \left(a\right)f\left(x\right)=0 \\ \left(b\right)f\left(x\right)>0 \\ \left(c\right)f\left(x\right)\le 0 \end{gathered}$$

(d) determine f

Short answer

(a)The function is 0 at $x=-3,2$,

(b)$f\left(x\right)>0atx>-3andx>2$,

(c)$f\left(x\right)\le 0atx\le -3andx=2$

(d)$f\left(x\right)=x^3-x^2-8x+12$

Step-by-step solution

Step 1. Given Information

The given graph of the function is

Part(a) Step 1. Explanation

Locate the points on the graph where the function intersects the x-axis.

Thus, at$x=-3andx=2$the function is equal to 0.

Part(b) Step 1. Explanation

Determine the domain of the function for which the function is above the x-axis.

At $x>-3andx>2$the function is strictly greater than 0.

In the interval notation,$x:(-3,\infty )\cup (2,\infty )$

Part(c) Step 1. Explanation

Determine the domain of the function for which the function is below or intersecting the x-axis.

At$x\le -3andx=2$and in interval notation the solution is$x:(-\infty ,-3]\cup \left\{2\right\}$

Part(d) Step 1. Calculation

Use the zeroes from part (a) to produce a polynomial. Expand the linear factors.

$$\begin{gathered} f\left(x\right)=(x+3)(x-2)(x-2) \\ =(x+3)(x^2-4x+4) \\ =x^3-4x^2+4x+3x^2-12x+12 \\ =x^3-x^2-8x+12 \end{gathered}$$