Chapter 1.5, Problem 12E

To determine

To check: Whether the function has an inverse function or not.

Answer to Problem 12E

The function $y=2^{3-x}$ has an inverse function.

Explanation of Solution

Given information: The function is $y=2^{3-x}$ .

Calculation:

Every function has an inverse function if it is one-to-one.

Use the graphing calculator to draw the graph of the function $y=2^{3-x}$ . Follow the steps written below:

• First press “ON” button to start graphical calculator.
• Press $\boxed{\text{Y}=}$ key and enter the right hand side of the given function after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2}\boxed{^}\boxed{(}\boxed{3}\boxed{-}\boxed{\text{X}}\boxed{)}$ .
• The display will show the equation:

  ${\text{Y}}_1=2^{3-\text{X}}$

• Press $\boxed{\text{GRAPH}}$ and then $\boxed{\text{TRACE}}$ key to produce the graph of the limit in the standard window.

  

From the graph, it can be observed that every horizontal line will intersect the curve only once. So, the function is one-to-one and if the function is one-to-one then it has an inverse function.

Therefore, the function $y=2^{3-x}$ has an inverse function.