Suppose we have a function
f(x) = 2x
So, if we input 2, we get 4 back
In Inverse of f ,
the opposite happens
i.e. we input 4 and we get 2
How to find the inverse?
f(x) = 2x
We put f(x) = y and find x in terms of y
y = 2x
y/2 = x
x = y/2
∴ f -1 (y) = y/2
Now,
f(f -1 (x)) will always give back x
i.e. f(f -1 (x)) is an identity function
Let’s check
f -1 (y) = y/2
So, f -1 (x) = x/2
f(f -1 (x)) = f(x/2)
= 2 (x/2)
= x
Similarly,
f -1 (f(x)) will always give back x
i.e. f -1 (f(x)) is an identity function
Let’s check
f -1 (f(x)) = f -1 (2x)
= 2x/2
= x
Thus,
f(f -1 (x)) and f -1 (f(x)) are identity functions .
Also, function will have inverse only when it is one-one and onto.
Let's next see how to check if function has inverse.