Chapter 1, Problem 30RE

(a)

To determine

To find: The domain of the function $y=3^{2-x}+1$ .

Answer to Problem 30RE

The domain of the function $y=3^{2-x}+1$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=3^{2-x}+1$ .

Calculation:

The function $y=3^{2-x}+1$ is an exponent function and is defined for all the real values of $x$ .

Therefore, the domain of the function $y=3^{2-x}+1$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

(b)

To determine

To find: The range of the function $y=3^{2-x}+1$ .

Answer to Problem 30RE

The range of the function $y=3^{2-x}+1$ is the interval $\left(1,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=3^{2-x}+1$ .

Calculation:

The value of the function $y=3^{2-x}+1$ is always greater than $1$ for all the real values of $x$ .

Therefore, the range of the function $y=3^{2-x}+1$ is the interval $\left(1,\infty \right)$ .

(c)

To determine

To graph: The function $y=3^{2-x}+1$ .

Explanation of Solution

Given information:

The given functionis $y=3^{2-x}+1$ .

Graph:

To graph a function $y=3^{2-x}+1$ , 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 $y=3^{2-x}+1$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{3}\boxed{^}\boxed{(}\boxed{2}\boxed{-}\boxed{\text{X}}\boxed{)}\boxed{+}\boxed{1}$ .

The display will show the equation,

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

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph of the function $y=3^{2-x}+1$ forms a curve in the Quadrant I.