Chapter 1.1, Problem 44E

To determine

To find: The values of $f\left(-3\right),f\left(0\right),f\left(2\right)$ and sketch the graph of piecewise function.

Answer to Problem 44E

The value of $f\left(-3\right)=\frac{9}{2}$.

The value of $f\left(0\right)=3$.

The value of $f\left(2\right)=-1$.

Explanation of Solution

Given:

The function is $f\left(x\right)=\{\begin{array}{l}3-\frac{1}{2}x\text{if }x<2\\ 2x-5\text{if }x\ge 2\end{array}$.

Calculation:

Obtain the values of $f\left(-3\right),f\left(0\right)\text{ and }f\left(2\right)$ as follows:

Since the value of x $\left(=-3\right)$ is less than 2, substitute −3 for x in the function $f\left(x\right)=3-\frac{1}{2}x$.

$\begin{array}{c}f\left(-3\right)=3-\frac{1}{2}\left(-3\right)\\ =3+\frac{3}{2}\\ =\frac{9}{2}\end{array}$

The value of $f\left(-3\right)=\frac{9}{2}$.

Since the value of x $\left(=0\right)$ is less than 2, substitute 0 for x in the function $f\left(x\right)=3-\frac{1}{2}x$.

$\begin{array}{c}f\left(0\right)=3-\frac{1}{2}\left(0\right)\\ =3\end{array}$

The value of $f\left(0\right)=3$.

Since the value of x is equal to 2, substitute 2 for x in the function $f\left(x\right)=2x-5$.

$\begin{array}{c}f\left(2\right)=2\left(2\right)-5\\ =4-5\\ =-1\end{array}$

The value of $f\left(2\right)=-1$.

When $x<2$, $f\left(x\right)=3-\frac{1}{2}x$ is a line with slope $-\frac{1}{2}$ and y-intercept 3 that lies to the left of the line x = 2.

When $x\ge 2$, $f\left(x\right)=2x-5$ is a line with slope 2 and y-intercept −5 that lies to the right of the line x = 2.

The graph of $f\left(x\right)$ is shown below in Figure 1.

In Figure 1, the open dot indicates that the point (2, 2) is excluded and the closed dot indicates that the point (2, −1) is included.