Inverse Of A Function

A function \( f: A \rightarrow B \) is said to have an inverse if there exists a function \(h : B \rightarrow A\) such that \(hof = I_A\) and \( foh = I_B \). The function h is called the inverse of f.

Inverse Of A Function

Inverse of a function f is denoted by \( f^{-1} \)

Conditions for the function ‘f’ to be invertible are

1) f must be a one-one function

2) f must be an onto function.

Let us go through few problems to understand the concept better.

Example 1: Consider \( f:{a, e, i, o, u} \rightarrow {a, b, c, d}\)  with \( f \) = \( {(a, a)(e, b)(i, c)(o, d)(u, d)}\). Check whether this function has an inverse.

Solution: The function is not one-one as two elements of the domain (o,u) are mapped to the same element (d) in the co-domain.

The function is onto as all the elements of the co-domain have pre-images (Note that if a function is not one-one, a check for onto is not necessary).

 The function f is not invertible as f is not one-one.

Example 2: Find the inverse of f if \( f: N \rightarrow R \) defined as \( f(x) \) = \(4x^2 + 20x + 25\) and \(f : N \rightarrow A \)  where A is the range of f, is invertible.

Solution:  Consider arbitrary element \(a ∈ A\).

\( a \) = \(4x^2 + 20x + 25 \) for some x in N

\( a \) = \((2x + 5)^2\)

\( 2x \) = \(\pm \sqrt{a} – 5\)

When \( -\sqrt{a} \) is considered, x becomes a negative number and hence is not a Natural number anymore. So,
\( 2x \) = \( \sqrt{a} – 5\)

\( x \) = \(\frac{(\sqrt{a} – 5)}{2}\)

\( f^{-1} \) = \(\frac{(\sqrt{a} – 5)}{2}\)

\( f^{-1}(x) \) = \( \frac{(\sqrt{x} – 5)}{2}\)

Example 3: Does an identity function with domain and co-domain as natural numbers have an inverse?

Solution: Consider the identity function f(x) = x

Identity function \(f: N \rightarrow N\)  is one-one and onto.

Therefore, f has an inverse.

Consider \(a ∈ N\) (co-domain)

\( a \) = \(x\)

\(f^{-1}\) = \( a \)

\(f^{-1}(x)\) =\( x\)

Example 4: Consider A = {1, 4, 9, 16, 25, 36}. Determine whether the function \( f: A \rightarrow A \) defined by f = {(1,1), (4, 9), (9, 16), (16, 4), (25, 36), (36, 25)} has an inverse.

Solution:  The function \( f: A \rightarrow A \) is one-one and onto. Therefore, the function has an inverse.

\(f^{-1}(1)\) = \( 1\)

\(f^{-1}(9)\) = \(4\)

\(f^{-1}(16)\) = \(9\)

\(f^{-1}(4)\) = \(16\)

\(f^{-1}(36)\) =\( 25\)

\(f^{-1}(25)\) = \(36\)<

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Practise This Question

What's the variable in the expression 5x+6?