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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)) areidentity 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.