Inverse Functions And Logarithms

 Inverse Functions

How to find the inverse of  functions

firstly we have to know what one-to-one means

A function f is called a one-to-one function if it never takes on the same value twice; that is,

\[f({x_1}) \ne f({x_2})\,\,\,whenever\,\,{x_1} \ne {x_2}\]

to explain it clearly

and we can check the graph if it's a one-to-one function easily by

For example:

 is one-to-one function

on the other hand 

 is not one-to-one function because the line cuts it in more than one point

The main idea of this lesson

Let f be a one-to-one function with domain A and range B.

Then its inverse function has domain B and range A and is defined by

\[{f^{ - 1}}(y) = x \Leftrightarrow f(x) = y\]

 for any y in B

to make this theorem easier just follow these steps

if your function is one-to-one just replace every y with x and every x with y and rewrite your equation with respect to y after that the domain will become range and range will become a domain

For example:

find the inverse of  

solution:

is this function one-to-one? yes it is we checked that before

so our functions is \[y = {x^3}\]  

\[domain = R  \&\ range = R \]

replace every y with x and every x with y

and the inverse of our functions is \[x = {y^3}\] 

old domain will become range and old range will become domain so

\[domain = R  \&\ range = R \]

rewrite the equation with respect to y \[y = {x^{1/3}}\]

another example:

find the inverse for 

is this function is one-to-one? yes because every value in a domain have a different value in the range 

and we can check that with Horizontal Line Test

our functions is 

\[y = \ln (x)\]

\[domain = (0,\infty )\& range = R\]

replace every y with x and every x with y

so the inverse of our function is \[x = \ln (y)\]

\[domain = R\& range = (0,\infty )\]

rewrite the equation with respect to y

\[y = {e^x}\]

Note: The graph of the inverse is obtained by reflecting the graph of the function about y=x

continued in part 2