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

$$f^{-1}(y)=x\iff f(x)=y$$

 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 $$y=x^3$$  

$$domain=R\mathrm{\&}range=R$$

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

$$domain=R\mathrm{\&}range=R$$

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 

$$y=\ln (x)$$

$$domain=(0,\mathrm{\infty })\mathrm{\&}range=R$$

replace every y with x and every x with y

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

$$domain=R\mathrm{\&}range=(0,\mathrm{\infty })$$

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