Chapter 2.3, Problem 73E

Solution

To determine the equation of the function graphed among the options.

The function $f\left(x\right)=-x\left(x+2\right)\left(x-2\right)$ given in the option $B)$ is the appropriate choice of the function represented in the graph.

Given the graph of the function:

  

Explanation:

The function represented by the graph seems to have only three zeroes: $-2$ , $0$ and $2$ .

Since the graph of the function passes the $x$ -axis directly at each of its zeroes, all the zeroes of the function has the multiplicity $1$ .

Then, there are two choices left:

  $\begin{array}{l}A)f\left(x\right)=-x\left(x+2\right)\left(2-x\right)\\ B)f\left(x\right)=-x\left(x+2\right)\left(x-2\right)\end{array}$ .

Now, it is clear from the graph that $f\left(x\right)$ is positive when $x=1$ .

Substitute $x=1$ in the function given in the option $A)$ :

  $\begin{array}{l}f\left(x\right)=-x\left(x+2\right)\left(2-x\right)\\ f\left(1\right)=-1\cdot \left(1+2\right)\left(2-1\right)\\ f\left(1\right)=-3\end{array}$ .

That is, $f\left(1\right)$ is negative for the function $f\left(x\right)=-x\left(x+2\right)\left(2-x\right)$ given in the option $A)$ .

That is, the option $A)$ is not the appropriate choice.

Then, it follows that $f\left(x\right)=-x\left(x+2\right)\left(x-2\right)$ given in the option $B)$ is the appropriate choice.