Chapter 1, Problem 98RE

To determine

The domain of the function in which it is increasing and find inverse of the function.

Answer to Problem 98RE

Function is increasing in interval $\left(2,\infty \right)$ and inverse of the function is $y=x+2$

Explanation of Solution

Given:

Function is given;

  $f\left(x\right)=\left|x-2\right|$

Let us consider f and g are inverse function.

Therefore,

  $f\left(g\left(x\right)\right)=x$ , in its domain.

  $g\left(f\left(x\right)\right)=x$ ,in its domain.

Then,

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

And also,

  $\begin{array}{l}\left(domain\right)f=\left(range\right)f^{-1}\\ \left(range\right)f=\left(domain\right)f^{-1}\end{array}$

Now, $f\left(x\right)=\left|x-2\right|$

Graph of the function will be,

  

From the graph it is clear that value of the function is decreases in the interval $\left(-\infty ,2\right)$ therefore function is decreasing in this interval.

And value of the increases in interval $\left(2,\infty \right)$ therefore function is increasing in this interval.

Now, find the inverse of the function in interval $\left(2,\infty \right)$

Let $y=f\left(x\right)$

Therefore,

  $y=x-2$

Now replace x and y with each other.

  $\begin{array}{l}x=y-2\\ y=x+2\end{array}$

So inverse of the function is $y=x+2$