Chapter 2, Problem 55RE

To determine

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

Answer to Problem 55RE

The intercepts are $\left(-4,0\right),\left(0,0\right),\left(4,0\right)$ .

The domain is the set of all real numbers The intercepts are $\left(-4,0\right),\left(0,0\right),\left(4,0\right)$ , domain is the set of all real numbers, and range is $\left\{y|y\ge -4\right\}$ or the interval $\left[-4,\infty \right)$ .

Explanation of Solution

Given information:

Given function

  $F\left(x\right)=\left|x\right|-4$

Calculation:

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

  

Shift the graph vertically down 4 units to get the graph of the function $F\left(x\right)=\left|x\right|-4$ .

  

Intercepts are the points where the graph touches or crosses the coordinate axes. In $F\left(x\right)=\left|x\right|-4$ , the graph touches the coordinate axes at $\left(-4,0\right),\left(0,0\right),\left(4,0\right)$ .

So, the intercepts are $\left(-4,0\right),\left(0,0\right),\left(4,0\right)$ .

In the function, 4 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 $-4$ and $\infty$ , and for each number $y$ , there is at least one number $x$ in the domain.

So, the range of $F\left(x\right)$ is $\left\{y|y\ge -4\right\}$ or the interval $\left[-4,\infty \right)$ .

Therefore, the intercepts are $\left(-4,0\right),\left(0,0\right),\left(4,0\right)$ , domain is the set of all real numbers, and range is $\left\{y|y\ge -4\right\}$ or the interval $\left[-4,\infty \right)$ .