Question

Consider the function

$g\left(x\right)=\left\{\begin{array}{ll}2x+1& ifx<-1\\ x-4& ifx\ge -1\end{array}\right.$

(a) Graph the function.

(b) List the intercepts.

(c) Find $g(-5)$.

(d) Find $g\left(2\right)$.

Short answer

(a) Graph of function is

(b) the x-intercept is $(4,0)$ and the y-intercept is $(0,-4)$

(c) value of $g(-5)$is $-9$

(d) value of $g\left(2\right)$is $-2$

Step-by-step solution

Step 1. given data

The given function is

$g\left(x\right)=\left\{\begin{array}{ll}2x+1& ifx<-1\\ x-4& ifx\ge -1\end{array}\right.$

Step 2. Part (a)

Determine the different points for the graph of the function $g\left(x\right)=2x+1$for which$x<-1$

Determine the different points for the graph of the function $g\left(x\right)=x-4$for which$x>=-1$

Step 2. graph of the function

Plot points and connect them through a straight line for the graph of functions

Step 4. Part (b)

For thex-intercept,$y=0$

then x value of $g\left(x\right)=x-4$when$g\left(x\right)=0$is

$$\begin{gathered} g\left(x\right)=x-4 \\ 0=x-4 \\ x=4 \end{gathered}$$

so x-intercept is $(4,0)$

For they-intercept,$x=0$soyvalue of$g\left(x\right)=x-4$ is

$$\begin{gathered} g\left(x\right)=x-4 \\ g\left(x\right)=0-4 \\ g\left(x\right)=-4 \\ y=-4 \end{gathered}$$

y-intercept is$(0,-4)$

Step 5. Part (c)

$x=-5$is less than$-1$, so the function will be $g\left(x\right)=2x+1$

substitute $-5$ for x in the function

$$\begin{gathered} g\left(x\right)=2x+1 \\ g(-5)=2(-5)+1 \\ g(-5)=-10+1 \\ g(-5)=-9 \end{gathered}$$

Step 6. Part (d)

$x=2$is greater than$-1$so function will be$g\left(x\right)=x-4$

substitute$2$forxin the function

$$\begin{gathered} g\left(x\right)=x-4 \\ g\left(2\right)=2-4 \\ g\left(2\right)=-2 \end{gathered}$$