Chapter 1.2, Problem 49E

(a)

To determine

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

Explanation of Solution

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

Graph:

To graph a function $f\left(x\right)=-\left|3-x\right|+2$ , 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 $f\left(x\right)=-\left|3-x\right|+2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(-)}\boxed{\text{MATH}}\boxed{\Rightarrow }\boxed{1}\boxed{3}\boxed{-}\boxed{\text{X}}\boxed{)}\boxed{+}\boxed{2}$ .

The display will show the equation,

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

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 $f\left(x\right)=-\left|3-x\right|+2$ is the graph of absolute value function reflected across $x-$ axis. Also, shifted 3 units right and 2 units upwards.

(b)

To determine

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

Answer to Problem 49E

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

Explanation of Solution

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

Calculation:

It is observed from the graph of function $f\left(x\right)=-\left|3-x\right|+2$ plotted in part (a), the function $f\left(x\right)=-\left|3-x\right|+2$ is defined for all the real values of $x$ .

So, the domain of the function $f\left(x\right)=-\left|3-x\right|+2$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

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

(c)

To determine

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

Answer to Problem 49E

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

Explanation of Solution

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

Calculation:

Range of a function is the set of output values. From the graph of function $f\left(x\right)=-\left|3-x\right|+2$ plotted in part (a), it can be observed that the range of the function is the interval $\left(-\infty ,2\right]$ .

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