Chapter 1.5, Problem 36E

Solution

To Analyse: The basic functions that inverses of their own.

The basic functions that inverses of their own are $x$ and $\frac{1}{x}$ .

Given:

Determine the basic functions that inverse of their own.

Calculation:

The basic functions which inverses of their own are $x$ and $\frac{1}{x}$ , as if $f\left(x\right)=x$ and $g\left(x\right)=x$ then,

Apply the formula for $g\left(x\right)$

  $f\left(g\left(x\right)\right)=f\left(x\right)$

Replace $x$ by $x$ in the $f\left(x\right)$ function:

  $f\left(g\left(x\right)\right)=x$

Now, apply the formula for $f\left(x\right)$ ,

  $g\left(f\left(x\right)\right)=g\left(x\right)$

Replace $x$ by $x$ in the $g\left(x\right)$ function:

  $g\left(f\left(x\right)\right)=x$

Thus when $f\left(x\right)=x$ and $g\left(x\right)=x$ , $f\left(g\left(x\right)\right)=x$ and $g\left(f\left(x\right)\right)=x$ .

So $f\left(x\right)=x$ is the basic function which is its own inverse.

Also, if $f\left(x\right)=\frac{1}{x}$ and $g\left(x\right)=\frac{1}{x}$ then

Apply the formula for $g\left(x\right)$ :

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

Replace $x$ by $\left(\frac{1}{x}\right)$ :

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

Simplify the function:

  $f\left(g\left(x\right)\right)=x$

Also,

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

Replace $x$ by $\left(\frac{1}{x}\right)$ :

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

Simplify the function:

  $g\left(f\left(x\right)\right)=x$

Hence, when $f\left(x\right)=\frac{1}{x}$ and $g\left(x\right)=\frac{1}{x}$ , $f\left(g\left(x\right)\right)=x$ and $g\left(f\left(x\right)\right)=x$ .

So, $f\left(x\right)=\frac{1}{x}$ is the basic function which is its own inverse.