Chapter 10.4, Problem 60E

(a)

To determine

To graph the function.

Answer to Problem 60E

It has its own inverse

Explanation of Solution

Given information: $g(x)=-x$

Formula Used: Inverse Relation

Calculation:

If a relation contains the point $\left(x,y\right)$ then the Inverse Relation contains the point $\left(y,x\right)$ because the inverse relation interchange the x and y coordinates.

  $g(x)=-x$

As,

  $y=g(x)$

Now,

  $y=-x$

So,

  $x=-y$

Now,

Inverse of $g\left(x\right)=-x$

So this function has its own inverse.

  

(b)

To determine

To graph the linear function.

Answer to Problem 60E

  $f\left(x\right)=\frac{1}{x}$

Explanation of Solution

Given information: function that has its own inverse

Formula Used: Inverse Relation

Calculation:

If a relation contains the point $\left(x,y\right)$ then the Inverse Relation contains the point $\left(y,x\right)$ because the inverse relation interchange the x and y coordinates.

Let,

  $f\left(x\right)=\frac{1}{x}$

As,

  $y=f(x)$

Now,

  $y=\frac{1}{x}$

So,

  $x=\frac{1}{y}$

Now,

Inverse of $f$ is $g\left(x\right)=\frac{1}{x}$

  

(c)

To determine

To find the general equation.

Answer to Problem 60E

  $f^{-1}\left(f\left(x\right)\right)=x$

Explanation of Solution

Given information: linear functions that are their own inverses

Formula Used: Inverse Relation

Calculation:

  $f^{-1}\left(f\left(x\right)\right)=x$

Let,

  $f\left(x\right)=\frac{1}{x}$

So,

  $f^{-1}\left(x\right)=\frac{1}{x}$

Now,

  $f^{-1}\left(f\left(x\right)\right)=\frac{1}{\frac{1}{x}}$

Hence,

  $f^{-1}\left(f\left(x\right)\right)=x$