Chapter 2, Problem 1RE

To determine

Whether the statement “The range of $f\left(x\right)=x^2$ is the same as the range of $f\left(x\right)=\left|x\right|$.” is true or false with the help of graph.

Answer to Problem 1RE

The statement “The range of $f\left(x\right)=x^2$ is the same as the range of $f\left(x\right)=\left|x\right|$.” is $\underset{\_}{\text{true}}$.

Explanation of Solution

It is given that the range of $f\left(x\right)=x^2$ is same as the range of $f\left(x\right)=\left|x\right|$.

The graphs of both the functions $f\left(x\right)=x^2$ and $f\left(x\right)=\left|x\right|$ are drawn below.

Draw the graph of $f\left(x\right)=x^2$ as shown below in Figure 1.

Draw the graph of $f\left(x\right)=\left|x\right|$ as shown below in Figure 2.

From the graphs of $f\left(x\right)=x^2$ and $f\left(x\right)=\left|x\right|$ , it can be seen that both the graphs lie above the x-axis.

For every point in its domain both the graphs give the positive value of function hence the range of both the functions is $\left[0,\infty \right)$.

Thus, the statement “The range of $f\left(x\right)=x^2$ is the same as the range of $f\left(x\right)=\left|x\right|$.” is $\underset{\_}{\text{true}}$.