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.
