Chapter 3.7, Problem 8E

To determine

To Graph: The function $v(x)=\left|x-3\right|$ and to compare with the graph of $f(x)=\left|x\right|$ and to find its domain and range.

Answer to Problem 8E

Domain $\in (-\infty ,\infty )$

Range $\in (0,\infty )$

Explanation of Solution

The graph of $v(x)=\left|x-3\right|$ is given by

  

And the graph of $f(x)=\left|x\right|$ is given by

  

Here the vertex of $f(x)=\left|x\right|$ is (0,0). Then to find vertex of $v(x)=\left|x-3\right|$ equate $\left|x-3\right|$ to 0

  $\begin{array}{l}\left|x-3\right|=0\\ \Rightarrow x=3\end{array}$

Then the vertex of v(x) is (3, 0)

So the vertex of v(x) is sifted to (3, 0) as compare to f(x).

And also in function v(x), x can take any possible real value

Then its domain is $(-\infty ,\infty )$

Thevalue of $\left|x-3\right|$ is always positive so the minimum value is $\left|x-3\right|$ is 0 at x=3

Then its range $\in (0,\infty )$