Inverse Functions And Logarithms

Published On :2021-10-16 05:18:00

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 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:
f(x) = {x^3} is one-to-one function
on the other hand 
f(x) = {x^2} 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  f(x) = {x^3}
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 f(x) = ln (x)

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