Inverse Functions

Algebra Level 3

How many functions are such that its inverse is itself?

That is, f ( x ) = f − 1 ( x ) f\left( x \right) =f^{ -1 }\left( x \right)

1000 ∞ \infty 64 2 3 1 0 4

Mateo Matijasevick by Mateo Matijasevick , 22, Colombia

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4 solutions

Kushal Bose
Jan 17, 2019

OK let me give Geometric explanations - A function f ( x ) f(x) and f − 1 ( x ) f^{-1}(x) are mirror image with respect to y = x y = x straight line . So , we can create infinite such curves in Euclidean plane which satisfies the above property . So. there exists infinitely such functions .

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Romeo Gomez
Sep 25, 2017

f ( x ) = 1 k x f(x)=\frac{1}{kx} for any k ≠ 0 k\neq 0 .

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Let A A to be any set . Take f : A ⟶ A , / f ( x ) = x = f − 1 ( x ) , ∀ x ∈ A f:A \longrightarrow A, \space / \space f(x) = x = f^{-1} (x), \space \forall x \in A . Since there are infinte sets these functions work.

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f ( x ) = a − x f(x)=a-x for any a, this will work.

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Good! All the perpendicular lines to y = x y=x . Also f ( x ) = k / x f(x)=k/x , with k ≠ 0 k\neq 0 , works.

Mateo Matijasevick - 5 years, 1 month ago

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