Chapter 3.1, Problem 40E

To determine

To find: The domain and range of the function $y=\left|x\right|$ .

Answer to Problem 40E

  $\begin{array}{l}\text{Domain}=\left(-\infty ,\infty \right)\\ \text{Range}=\left[0,\infty \right)\end{array}$

Explanation of Solution

Given information:

Given function is $y=\left|x\right|$

Concept used:

Domain: - The domain of the function is the set of input for which the given function real and defined.

Range: - The set of value of the dependent variable for which a function is defined.

Since, the range of an absolute function of the form $c\left|ax+b\right|+k$ is $f\left(x\right)\ge k$

Calculation:

First find the domain of the function $y=\left|x\right|$

Since the function $y=\left|x\right|$ has no undefined point therefore domain is $-\infty <x<\infty$ .

In interval notation domain is $\left(-\infty ,\infty \right)$

Now find the range of the function $y=\left|x\right|$

The range of an absolute function of the form $c\left|ax+b\right|+k$ is $f\left(x\right)\ge k$

Here, $k=0$

Hence, range is $0\le x<\infty$

In interval notation range is $\left[0,\infty \right)$ .

Conclusion:

  $\begin{array}{l}\text{Domain}=\left(-\infty ,\infty \right)\\ \text{Range}=\left[0,\infty \right)\end{array}$