Chapter 2.3, Problem 51E

To determine

To explain: The reason that the equation $e^{-x}=x$ has at least one solution.

Answer to Problem 51E

Yes, the equation $e^{-x}=x$ has one solution as graph of both the functions intersect at one point.

Explanation of Solution

Given information: The equation is $e^{-x}=x$.

Solve the equation $e^{-x}=x$ with the help of graph.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter left hand side of the equation $e^{-x}=x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{LN}}\boxed{(-)}\boxed{\text{X}}\boxed{)}$ . Now, press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $e^{-x}=x$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{\text{X}}$.

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=e^\left(-\text{X}\right)\\ {\text{Y}}_2=\text{X}\end{array}$

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)

As observed from the graph, both the graphs intersect at only one point.

Therefore, the equation $e^{-x}=x$ has one solution as graph of both the functions intersect at one point.