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
FAQ Related to Inverse of a Function
What is the formula for inverse?
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.
Why do we use inverse functions?
Inverse processes are necessary to solve equations because they allow reversing mathematical operations such as logarithms, inverses of exponential functions, solving exponential equations, etc.
What is the symbol for the inverse of a function?
The symbol of the inverse function depends on the symbol of the function such that if the function is denoted by f then the inverse function will be f ⁻¹.
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.