For each pair of inverse functions, show that (f ∘f^-1)(x)=x...

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Key Concepts

Inverse Functions

Inverse functions are pairs of functions that undo each other’s operations. When a function f and its inverse f?¹ are composed in either order—f followed by f?¹ or f?¹ followed by f—the result is the original input value. This relationship is defined by the equations f(f?¹(x)) = x and f?¹(f(x)) = x, ensuring that one function exactly reverses the effect of the other.

Composition of Functions

Composition of functions involves applying one function to the result of another function. This process is fundamental when working with inverse functions because the composition must yield the identity function. Composing f with its inverse demonstrates that the operations cancel out, leading to an input-preserving outcome.

Identity Function

The identity function is a function that returns its input unchanged, typically expressed as I(x) = x. In the context of inverse functions, when a function f is composed with its inverse f?¹, the result is the identity function. Proving that f(f?¹(x)) = x and f?¹(f(x)) = x confirms that the functions effectively undo each other.

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