# Inverse Of A Function

If f: A -> B is a one-one onto (or bijective) function and if f(x) = y, where x
A, y
B then f1 : B -> A defined
by f1 (y) = x, is known
as inverse function of f.

A function whose inverse exists (i.e. if it is bijective) is called an

A function whose inverse exists (i.e. if it is bijective) is called an

**Invertible function**.**For example:**- If A = {2, 4, 6} and B = {5, 7, 9} and f: A -> B is given by f(2) = 5, f(4) = 7
and f(6) = 9.

Find f 1.

**Solution:**

Given that f(2) = 5, f(4) = 7 and f(6) = 9,

Hence f = {(2,5), (4, 7), (6, 9)}

Hence f1 = {(5, 2), (7, 4), (9, 6)}

- Let f: N -> R defined as f(x) = x2
+ 4x + 8, then show that f: N -> A, where A is the range of f, is invertible. Also
find f1.

Solution: f(x1) = f(x2) => x12 + 4 x1 + 8 = x22 + 4 x2 + 8

=> (x12 – x22) + 4(x1 – x2) = 0

=> x1 = x2

Hence f is**one-one**.

Let y A be any element then f(x) = y

=> x2 + 4x + 8 = y

=> y = (x + 2)2 + 4

=> x = (√y - 4 – 2), x R

And f(√y - 4 – 2) = (√y - 4 – 2)2 + 4(√y - 4 – 2) + 8 = y => f is invertible.

Hence f1: A -> N can be defined as f1(x) = (√x - 4 – 2).

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