Chapter 7.7, Problem 57PPS

(a)

To determine

To graph: The function and their axis of symmetry $y=xanda\left(x\right)=e^{-x}$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=e^{x}andg\left(x\right)=\ln x$ .

Graph:

Calculate the function.

The graph of the Plot $f\left(x\right),g\left(x\right),a\left(x\right)$ and the axis of symmetry in the same graph using maple for $-5\le x\le 5$ .

  

Figure (1)

Interpretation: Graph for the functions shows in figure (1).

(b)

To determine

To find: The reflection of $f\left(x\right)$ along the axis.

Answer to Problem 57PPS

The reflection of $f\left(x\right)=e^x$ t is $x$ axis is $a\left(x\right)=-e^x$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=e^x$ and $g\left(x\right)=\ln x$ .

Calculation:

Calculate the function.

Plot $f\left(x\right)=e^{x}anda\left(x\right)=e^{-x}$ are the reflection of each other with respect to the y axis

Therefore the reflection of $f\left(x\right)=e^x$ about the other axis that is $x$ axis is $a\left(x\right)=-e^x$ .

(c)

To determine

To graph : The two function that are reflection of $g\left(x\right)$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=e^x$ and $g\left(x\right)=\ln x$ .

Graph:

Calculate the function.

The reflection of $g\left(x\right)=\ln x$ about the other axis that $x$ axis is $c\left(x\right)=-\ln x$ .

The reflection of $g\left(x\right)=\ln x$ about the other axis that $y$ axis is $d\left(x\right)=-\ln x,x<0$ .

Plot The function $c\left(x\right),d\left(x\right)$ in the same graph using maple for $-5\le x\le 5$ .

  

Figure (1)

Interpretation: Graph for The function $c\left(x\right),d\left(x\right)$ in the same graph using maple for $-5\le x\le 5$ shown is figure (1) .

(d)

To determine

To find: The functions that we have graphed inverses as well.

Answer to Problem 57PPS

They must have unique inverse.

Explanation of Solution

Given information:

The given function is $f\left(x\right)=e^x$ and $g\left(x\right)=\ln x$ ..

Calculation:

Calculate the function.

There are no function except $f\left(x\right),g\left(x\right)$ which are inverse of $g\left(x\right),f\left(x\right)$ function are the reflection about $y=-x$ .the function $f\left(x\right),g\left(x\right)$ are function.

Therefore they must have unique inverse.