Chapter 5.1, Problem 123E

To determine

To find: inverse of the given function, if exist.

Answer to Problem 123E

The function has its inverse.

  $f^{-1}\left(x\right)=\frac{x^2+14}{3}$

Explanation of Solution

Given information :

A function is given as:

  $f\left(x\right)=\sqrt{3x-14}$

Concept used:

A function has an inverse if the function is one to one and onto.

A vertical line should intersect the function once.

Inverse of any function can be found by considering $y=f\left(x\right)$ and interchanging $x$ by $y$ in thefunction, then expressing $y$ in terms of $x$ .

Calculation :

Consider the given function.

  $f\left(x\right)=\sqrt{3x-14}$

Let’s apply vertical line test on the function as shown:

  

Here, the vertical line $x=10$ intersects the function once.

Therefore, the function $f\left(x\right)=\sqrt{3x-14}$ has an inverse.

Now, evaluate the inverse as shown:

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

So,

  $y=\sqrt{3x-14}$

Replace $x$ by $y$ .

  $x=\sqrt{3y-14}$

Now, express $y$ in terms of $x$ .

  $\begin{array}{c}x=\sqrt{3y-14}\\ x^2=3y-14\\ 3y=x^2+14\\ y=\frac{x^2+14}{3}\end{array}$

Hence, $f^{-1}\left(x\right)=\frac{x^2+14}{3}$ .