Chapter 1, Problem 93RE

To determine

Domain of function in which is increasing and $f^{-1}$ of the function.

Answer to Problem 93RE

Domain of the function for which function is increasing is $\left(0,\infty \right)$ and inverse of the function is $\sqrt{\frac{x}{2}}+4$

Explanation of Solution

Given:

Function is given as;

  $f\left(x\right)=2{\left(x-4\right)}^2$

There are mainly two methods for finding the interval in which function is increases or decreases one is by graphically and one is by calculus method.

Other way to find the interval is by comparing with parent function different region can be estimated.

The function which is given is a transform of squaring function is increasing for all values greater than 0.

Therefore,

  $\begin{array}{l}f\left(x\right)\ge 0\\ 2{\left(x-4\right)}^2\ge 0\\ x\ge 0\end{array}$

Now, let $y=f\left(x\right)$

  $\begin{array}{l}y=2{\left(x-4\right)}^2\\ \left(x-4\right)=\sqrt{\frac{y}{2}}\\ x=\sqrt{\frac{y}{2}}+4\end{array}$

Now replacing x and y with each other.

  $y=\sqrt{\frac{x}{2}}+4$

Therefore inverse of the is $y=\sqrt{\frac{x}{2}}+4$