Determine whether the given functions are inverses.

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If a function f takes an input x and produces an output y, then its inverse function f⁻¹ takes y back to x. For two functions to be inverses, applying one after the other should yield the original input, meaning f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in their domains.

Composition of Functions

The composition of functions involves combining two functions to create a new function. If f and g are two functions, their composition is denoted as (f ∘ g)(x) = f(g(x)). To determine if two functions are inverses, we need to evaluate their compositions in both orders and check if they equal the identity function, which is simply x.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. When determining if two functions are inverses, it is crucial to consider their domains and ranges, as they must align appropriately. Specifically, the range of the original function must match the domain of its inverse, and vice versa, for the functions to be valid inverses.