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Here in this post , we will see how to find the inverse of a function
Inverse Function
- A function can have inverse only if it is bijective
- Note that not all functions have inverses.
- Sometime , the function may not be bijective from R -> R, you can restrict the domain and range of the function to make it one-to-one and onto and then find the inverse of the restricted function.
how to find the inverse of a function
Method I
- let f(x) be the function f(x)=4x -2
- Consider an arbitrary element y then by definition y=4x-2
- Find out the value of x in terms of y
$y=4x-2$
$y+2=4x$
$x= \frac {y+2}{4}$ - Now the invertible function can be defined as
$g(x) = \frac {x+2}{4}$
Method II
- let f(x) be the function f(x)=4x -2
- Consider an arbitrary element y then by definition y=4x-2
- Swap the position of x and y
$y=4x-2$
$x=4y -2$
$y= \frac {x+2}{4}$ - Now the invertible function can be defined as
$g(x) = \frac {x+2}{4}$
Method III
Let f(x) be the function and $f^{-1}(x) $ be the inverse
We can use below equation
$f(f^{-1}(x)) =x$
Example
f(x)=4x -2
$f(f^{-1}(x)) =x$
or
$4 f^2{-1}(x) -2 =x$
$f^2{-1}(x) = \frac {x+2}{4}$
Solved Examples
Example 1
Let f: R -> R given as $f(x) =x^3 + 1$, find the inverse $f^{-1}(x)$
Solution
let $y=x^3 + 1$
or $ x^3 =1 -y$
or $x = (1-y)^{1/3}$
So inverse function is
$f^{-1}(x)= (1-x)^{1/3}$
Example 2
Let f: R ->R given as $f(x) = 4 – (x–7)^3$. Find the inverse $f^{-1}(x)$
Solution
Let $y= 4 – (x–7)^3$
$(x-7)^3=4-y$
$x= (4-y)^{1/3} + 7$
So inverse function is
$f^{-1}(x)= (4-x)^{1/3} + 7$
Example 3
if f: [2,\infty)$ -> ($-\infty$,4] given as f(x)=x(4-x). Find the inverse $f^{-1}(x)$
Solution
Let $y= x(4-x)$
$x^2 -4x + y=0$
$x= \frac {4 \pm \sqrt {16-4y}}{2}$
$x= 2 \pm \sqrt {4-y}$
as $x \geq 2$
$x= 2 + \sqrt {4-y}$
So inverse function is
$f^{-1}(x)= 2 + \sqrt {4-x}$