Q. 54 - Precalculus Enhanced with Graphing Utilities

Question

The function $\mathit{f}$ is one-to-one. Find its inverse and check your answer. Graph $\mathit{f}$, ${\mathit{f}}^{\mathbf{-}\mathbf{1}}$, and y = x on the same coordinate axes.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}$

Step-by-Step Solution

Answer

The inverse of the function is ${\mathit{f}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{x}\mathbf{-}\mathbf{1}}$.

Step 1. Given information :

The given function is one-to-one function.

$\mathit{f}\left(x\right)\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}$.

Step 2. Finding the inverse of the function :

For finding the inverse of the function we need to interchange the variables.

let, $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}$ 

Interchanging the variables x, y,

$$\begin{gathered} \mathit{x}\mathbf{=}{\mathit{y}}^{\mathbf{3}}\mathbf{+}\mathbf{1} \\ {\mathit{y}}^{\mathbf{3}}\mathbf{=}\mathit{x}\mathbf{-}\mathbf{1} \\ \mathit{y}\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{x}\mathbf{-}\mathbf{1}} \end{gathered}$$

So, the inverse ${\mathit{f}}^{\mathbf{-}\mathbf{1}}\left(x\right)\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{x}\mathbf{-}\mathbf{1}}$.

Step 3. Check the solution :

To check whether the function is inverses are not,

let, $\mathit{f}\left(f^{-1}\left(x\right)\right)\mathbf{=}\mathit{x}$

$$\begin{gathered} \mathit{f}\left(\sqrt[3]{x-1}\right)\mathbf{=}\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{w}\mathit{e}\mathbf{ }\mathit{g}\mathit{o}\mathit{t}\mathbf{,}\mathbf{ }\mathbf{ }{\mathit{f}}^{\mathbf{-}\mathbf{1}}\left(x\right)\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{x}\mathbf{-}\mathbf{1}} \\ {\left(\sqrt[3]{x-1}\right)}^{\mathbf{3}}\mathbf{+}\mathbf{1}\mathbf{=}\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{g}\mathit{i}\mathit{v}\mathit{e}\mathit{n}\mathbf{,}\mathbf{ }\mathbf{ }\mathit{f}\left(x\right)\mathbf{=}\mathbf{ }{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1} \\ \mathbf{(}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{+}\mathbf{1}\mathbf{=}\mathit{x} \\ \mathbf{ }\mathit{x}\mathbf{=}\mathit{x} \end{gathered}$$

Similarly, ${\mathit{f}}^{\mathbf{-}\mathbf{1}}\left(f\left(x\right)\right)\mathbf{=}\mathit{x}$

$$\begin{gathered} {\mathit{f}}^{\mathbf{-}\mathbf{1}}\mathbf{(}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{g}\mathit{i}\mathit{v}\mathit{e}\mathit{n}\mathbf{,}\mathbf{ }\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1} \\ \sqrt[\mathbf{3}]{\mathbf{(}{\mathbf{x}}^{\mathbf{3}}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{ }\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{,}\mathbf{ }{\mathit{f}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{x}\mathbf{-}\mathbf{1}} \\ \sqrt[\mathbf{3}]{{\mathbf{x}}^{\mathbf{3}}}\mathbf{ }\mathbf{=}\mathbf{ }\mathit{x} \\ x\mathbf{=}\mathit{x}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{,}\mathbf{ }{\mathit{f}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)} \\ \mathit{h}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{ }\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{d}\mathbf{ }\mathbf{.} \end{gathered}$$

Step 4. Graph representation of f , f - 1   a n d   y = x :

Graph ;

Domain of $\mathit{f}$= range of ${\mathit{f}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$

Domain of ${\mathit{f}}^{\mathbf{-}\mathbf{1}}$ = range of $\mathit{f}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.