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Inverse Of A Function

Inverse of a Function Online Tutoring / Homework Help
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 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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