What does the inverse tell you?

An inverse function plays a crucial role in mathematics by retrieving the original input value from the output generated by another function. When two functions, f and g, are inverses of each other, a unique property emerges.

Specifically, applying function f to the output of function g, or vice versa, results in the identity function. This means that f(g(x)) equals x and g(f(x)) also equals x. This property defines the inverse relationship between f and g.

To illustrate, consider a function f(x) and its supposed inverse g(y). If we know the expression for g(y) in terms of y and want to verify if it is indeed the inverse of f(x), we can use the above property.

For instance, let's assume g(y) is given by (y-5)/2 = x. To determine if g(y) is the inverse of some function f(x), we would need to find f(x) such that f(g(y)) = y. Alternatively, expressing g(y) in terms of x and then finding f(x) that satisfies f(g(x)) = x would confirm the inverse relationship.

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