Chapter 2, Problem 57RE

To determine

To sketch: the graph of the function. Also state the domain, intercepts and range of the graph.

Answer to Problem 57RE

The intercepts are $\left(0,0\right)$ .

The domain is the set of all real numbers.

The intercepts are $\left(0,0\right)$ , domain is the set of all real numbers, and range is $\left\{y|y\ge -\infty \right\}$ or the interval $\left[-\infty ,0\right]$ .

Explanation of Solution

Given information:

Given function

  $g\left(x\right)=-2\left|x\right|$

Calculation:

Graph the basic function $g\left(x\right)=\left|x\right|$ .

  

Reflect the graph $g\left(x\right)=2\left|x\right|$ is obtained by multiplying each $y$ coordinate of $g\left(x\right)=\left|x\right|$ by 2.

  

Reflect the graph $g\left(x\right)=2\left|x\right|$ about the $x$ axis to get the graph of the function $g\left(x\right)=-2\left|x\right|$ .

  

Intercepts are the points where the graph touches or crosses the coordinate axes. In $F\left(x\right)=-2\left|x\right|$ , the graph touches the coordinate axes at $\left(0,0\right)$ .

So, the intercepts are $\left(0,0\right)$ .

In the function, 2 is subtract from $\left|x\right|$ . Since, these operations can be performed on any real number; the domain is the set of all real numbers.

The points on the graph have $y$ coordinates between $-\infty$ and 0, and for each number $y$ , there is at least one number $x$ in the domain.

So, the range of $F\left(x\right)$ is $\left\{y|y\ge -\infty \right\}$ or the interval $\left[-\infty ,0\right]$ .

Therefore, the intercepts are $\left(0,0\right)$ , domain is the set of all real numbers, and range is $\left\{y|y\ge -\infty \right\}$ or the interval $\left[-\infty ,0\right]$ .