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

Key Concepts

Inverse Functions

Inverse functions are functions that 'undo' one another. If f is a function, then its inverse function f?¹ satisfies the condition that applying f followed by f?¹, or vice versa, returns the original input. This means that for all x in the domain of f?¹, f(f?¹(x)) = x, and for all x in the domain of f, f?¹(f(x)) = x.

Function Composition

Function composition is the operation of applying one function to the results of another. In the context of inverse functions, composing a function with its inverse, whether as f(f?¹(x)) or f?¹(f(x)), should yield the identity function on appropriate domains. This process is fundamental in verifying that two functions are inverses of each other.

Identity Function

The identity function is a function that always returns its input, i.e., I(x) = x. When a function is composed with its inverse, the result is the identity function on the respective domain. Verifying that the composition equals x demonstrates that the functions indeed reverse each other’s effect.

Domains and Ranges

Understanding domains and ranges is crucial when dealing with inverse functions and their compositions. The domain of the original function must match the range of its inverse and vice versa. This compatibility ensures that the compositions make sense and that each step of the function’s application is valid, so that the identity property holds true.

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