Today you will learn "how to find the Inverse of a function".
Before knowing how to find the Inverse of a Function, we need to know what is the inverse of a function, after that we will understand their calculation process step by step with an example.
So, without wasting time let's get started.
What is the Inverse of a Function?
If f is a function then the set of ordered pairs obtained by interchanging the first and second coordinates of each ordered pair in f is called the inverse of f. It is denoted by f ⁻¹.
A function f has an inverse if and only if f is injective and f is injective and surjective.
Let f: A → B be a function. The inverse of f is another function f - ¹: B → A.
So,
x = f - ¹( y ) ⇔ y = f ( x )
How to Find the Inverse of a Function?
If the function f is given, then the inverse function f ⁻¹ can be found by an algebraic method that involves the following steps.
Step - 1
Write the function y = f ( x ) form .
Step - 2
Solve the equation in Step - 1 for x in terms of y.
Step - 3
In the resulting equation in Step - 2, replace x with f⁻¹ (y).
Step - 4
Replace each y in the result of Step - 3 by x to get f ⁻¹ ( x ).
Step - 5
Check the answer by verifying that
f ⁻¹ ( f ( x ) ) = x .
Example
Let f: R → R be the function defined by
f ( x ) = 2x - 7. find f ⁻¹ ( x )?
Solution
Given that,
f( x ) = 2x - 7 ,
we are to find f - ¹ ( x )
Step - 1
Write the function y = f(x)
So,
y = 2x - 7
Step - 2
Now solving the equation for x in terms of y so,
y = 2x - 7
2x = y + 7
Step - 3
Now replace the x with f ⁻¹(y)
So,
f ⁻¹(y) = (y +7)/2
Step - 4
Now replace each y by x to get f ⁻¹(x)
So,
f ⁻¹(x) = (x +7)/2
Step - 5
Now we will check the answer by verifying that
f ⁻¹ ( f ( x ) ) = x .
So,
We have f ( x ) = 2x - 7 and f ⁻¹(x) = (x +7)/2
Now,
f ⁻¹{f(x)} = f ⁻¹(2x - 7)
= (2x - 7 + 7)/2
= 2x/2
= x
Hence prove it,
f ⁻¹ ( f ( x ) ) = x
Question
If f : R → R be the function defined by
f ( x ) = (x - 3)/2. find f ⁻¹ ( x ) and also verify?
Solution
Let,
y = f(x)
So,
y = (x-3)/2
Swap x with y so,
x = (y -3)/2
2x = y - 3
y = 2x + 3
Since,
y = f ⁻¹(x)
So,
f ⁻¹(x) = 2x + 3
So here we explained step by step how to find the inverse of a function with an example if you are still confused then you can contact me via comment or via mail.
Thank You.
