Chapter 1.5, Problem 35E

Solution

To Analyse: The pair of basic functions that inverses of each other.

The basic functions which are inverses of each other are $e^x$ and $\ln x$ .

Given:

Determine the pair of basic functions that inverse with each other.

Calculation:

The basic functions which are inverses of each other are $e^x$ and $\ln x$ , as if $f\left(x\right)=e^x$ and $g\left(x\right)=\ln x$ then,

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

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

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

  $f\left(g\left(x\right)\right)=e^{\ln x}$

Simplify the function:

  $f\left(g\left(x\right)\right)=x$

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

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

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

  $g\left(f\left(x\right)\right)=\ln \left(e^x\right)$

Simplify the function:

  $g\left(f\left(x\right)\right)=x$

So, when $f\left(x\right)=e^x$ and $g\left(x\right)=\ln x,f\left(g\left(x\right)\right)=x$ and $g\left(f\left(x\right)\right)=x$

So, $e^x$ and $\ln x$ are inverse functions.

Another example:

Also, if the domain of the basic function $f\left(x\right)=x^2$ is restricted to $\left[0,\infty \right)$ then $g\left(x\right)=\sqrt{x}$ is the inverses of each other:

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

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

Replace $x$ by $\sqrt{x}$

  $f\left(g\left(x\right)\right)={\left(\sqrt{x}\right)}^2$

Simplify the function:

  $f\left(g\left(x\right)\right)=x$

Also,

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

  $g\left(f\left(x\right)\right)=g\left(x^2\right)$

Replace $x$ by $x^2$

  $g\left(f\left(x\right)\right)=\sqrt{\left(x^2\right)}$

Simplify the function:

  $g\left(f\left(x\right)\right)=x$

Thus when $f\left(x\right)=x^2$ , with domain $\left[0,\infty \right)$ and $g\left(x\right)=\sqrt{x},f\left(g\left(x\right)\right)=x$ and $g\left(f\left(x\right)\right)=x$ . So $x^2$ , with domain $\left[0,\infty \right)$ and $\sqrt{x}$ are inverse functions.