Chapter 2.5, Problem 59AYU

In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Stan with the graph of the basic function (for example, $y\text{ = }{x}^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

59. $g\left(x\right)\text{ = 2}\left|1\text{ - }x\right|$

To determine

To graph: The function $f\left(x\right)=2\left|1-x\right|$, using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, $y=x^2$) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.

Answer to Problem 59AYU

Domain of the function $f\left(x\right)=2\left|1-x\right|$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=2\left|1-x\right|$ is $\left[0,\infty \right)$.

Explanation of Solution

Given:

$f\left(x\right)=2\left|1-x\right|$

Graph:

Now use the following steps to obtain the graph of $f\left(x\right)$

Step 1: The function $f\left(x\right)=\left|x\right|$ is the absolute value function.

$f\left(x\right)=\left|x\right|$     absolute value function

Step 2: To obtain the graph of $f\left(x\right)=\left|x+1\right|$, replace $x$ by $x+1$ from each $y\text{-coordinate}$ on the graph of $f\left(x\right)=\left|x\right|$, that it is shifted left 1 unit.

$f\left(x\right)=\left|x+1\right|$     replace $x$ by $x+1$; horizontal shift left 1 unit.

Step 3: To obtain the graph of $f\left(x\right)=\left|1-x\right|$, replace $x$ by $-x$ of the graph of $f\left(x\right)=\left|x+1\right|$, that it is reflection about the $y\text{-axis}$.

$f\left(x\right)=\left|1-x\right|$     replace $x$ by $-x$; reflection about $y\text{-axis}$.

Step 4: To obtain the graph of $f\left(x\right)=2\left|1-x\right|$, multiply each $y\text{-coordinate}$ of the graph of $f\left(x\right)=\left|1-x\right|$ by 2, that it is vertically stretched by the factor of 2.

$f\left(x\right)=2\left|1-x\right|$     multiply by 2, vertically stretched by a factor of 2

Interpretation:

Domain of the function $f\left(x\right)=2\left|1-x\right|$ is $\left(-\infty ,\infty \right)$.

Range of the function $f\left(x\right)=2\left|1-x\right|$ is $\left[0,\infty \right)$.