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# Finding the Equation of an Inverse Function

Since f(x) and f -1(x) are inverses, they â€œundoâ€ each other. For example, the function f(x) = x + 2 is a rule that says â€œadd 2 to the input.â€ The inverse of this function is f -1(x) = x - 2. This is a rule that says â€œsubtract 2 from the input.â€

As a consequence, the composition of f(x) and f -1(x) simplifies to x.

Property â€” Composition of a Function and Its Inverse

If a function, f(x), has an inverse, f -1(x), then: for every x in the domain of f, and for every x in the domain of f -1.

Example 1

Given f(x) = 5x - 4 and , determine if g(x) is the inverse of f(x).

Solution

If g(x) is the inverse of f(x), then the composition (f g)(x) will equal x.

 Find (f ○ g)(x). (f ○ g)(x) = f[g(x)] Replace g(x) with  In f(x), replace x with  Cancel common factors of 5. Subtract. = x + 4 - 4= x

Since (f g)(x) = x, g(x) is the inverse of f(x).

Note:

We can also use (g f)(x) to see if is the inverse of f(x) = 5x - 4.