Chapter 3.1, Problem 41E

To determine

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

Answer to Problem 41E

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

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 $y=-\left|x\right|$

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

Since the function $y=-\left|x\right|$ has neither undefined point nor domain constraints therefore domain is $-\infty <x<\infty$ .

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

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

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

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

Here, $k=0$

Hence, range is $-\infty <f\left(x\right)\le 0$

In interval notation range is $\left(-\infty ,0\right]$ .