Q. 7 - Precalculus Enhanced with Graphing Utilities

Question

Consider the function

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

(a) Graph the function.

(b) List the intercepts.

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

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

Step-by-Step Solution

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 1. given data

The given function is 

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

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 the x-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 the y-intercept,$x=0$ so y value 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 $2$ for x in the function

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