Finding the Equation of an Inverse Function

Since f(x) and f ^-1(x) are inverses, they “undo” each other. For example, the function f(x) = x + 2 is a rule that says “add 2 to the input.” The inverse of this function is f ^-1(x) = x - 2. This is a rule that says “subtract 2 from the input.”

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

Property — 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

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.