Chapter 1.2, Problem 50E

(a)

To determine

To graph: The function $f\left(x\right)=2\left|x+4\right|-3$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=2\left|x+4\right|-3$ .

Graph:

To graph a function $f\left(x\right)=2\left|x+4\right|-3$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $f\left(x\right)=2\left|x+4\right|-3$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{\times }\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{1}\boxed{\text{X}}\boxed{+}\boxed{4}\boxed{)}\boxed{-}\boxed{3}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=2*\text{abs}\left(\text{X}+4\right)-3$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below:

  

Figure (1)

Interpretation: From the graph it can be observed that $f\left(x\right)=2\left|x+4\right|-3$ is the graph of absolute value function stretched vertically, shifted 4 unit to the left and 3 units downward.

(b)

To determine

To find: The domain of the function $f\left(x\right)=2\left|x+4\right|-3$ .

Answer to Problem 50E

The domain of the function $f\left(x\right)=2\left|x+4\right|-3$ is the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=2\left|x+4\right|-3$ .

Calculation:

Domain is the set of all input values of $x-$ values. The graph of function $f\left(x\right)=2\left|x+4\right|-3$ plotted in part (a), it can be observed that the function is defined for all the real values of $x$ .

So, the domain of the function is the set of real numbers.

Therefore, the domain of the function $f\left(x\right)=2\left|x+4\right|-3$ is the interval $\left(-\infty ,\infty \right)$ .

(c)

To determine

To find: The range of the function $f\left(x\right)=2\left|x+4\right|-3$ .

Answer to Problem 50E

The range of the function $f\left(x\right)=2\left|x+4\right|-3$ is the interval $\left[-3,\infty \right)$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=2\left|x+4\right|-3$ .

Calculation:

It is observed from the graph of function $f\left(x\right)=2\left|x+4\right|-3$ plotted in part (a), the function has the minimum value $f\left(x\right)=-3$ and no maximum value.

So, the range of the function is the interval $\left[-3,\infty \right)$ .

Therefore, the range of the function $f\left(x\right)=2\left|x+4\right|-3$ is the interval $\left[-3,\infty \right)$ .