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
Given f(x) = 5x - 4 and
determine if g(x) is the inverse
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)
|Replace g(x) with
|In f(x), replace x with
|Cancel common factors of 5.
||= x + 4 - 4
Since (f ○ g)(x) = x, g(x) is the
inverse of f(x).
We can also use (g ○ f)(x) to see if
is the inverse of f(x)
= 5x - 4.