Operations with Functions
Inverse Functions
Inverse Functions
Two functions f and g are inverse functions if f o g(x) = x and g o f (x) = x for all values of x in the domain of f and g .
For instance,
f (x) = 2x
and
g(x) = 
x
are inverse functions
because
f
o
g(x) = f (g(x)) = f (
x) = 2(
x) = x
and
g
o
f (x) = g(f (x)) = g(2x) = (2x) = x
. Similarly,
f (x) = x + 1
and
g(x) = x - 1
are inverse funcions because
f
o
g(x) = f (g(x)) = f (x - 1) = (x - 1) + 1 = x
and
g
o
f (x) = g(f (x)) = g(x + 1) = (x + 1) - 1 = x
.
h(x) = 3x - 1
and
j(x) = 
are inverse functions because
h
o
j(x) = h(j(x)) = h() = 3() - 1 = x + 1 - 1 = x
and
j
o
h(x) = j(h(x)) = j(3x - 1) = 
= 
= x
.
The inverse of a function f (x) is denoted f -1(x) .
Finding the Inverse of a Function by Reversing Operations
The trick to finding the inverse of a function f (x) is to "undo" all the operations on x in reverse order.
The function f (x) = 2x - 4 has two steps:
- Multiply by 2.
- Subtract 4.
- Add 4.
- Divide by 2.
.We can verify that this is the inverse of f (x) :
f -1(f (x)) = f -1(2x - 4) ==
= x .
f (f -1(x)) = f (
Example 1: Find the inverse of
f (x) = 3(x - 5)
.
Original function:
- Subtract 5.
- Multiply by 3.
- Divide by 3.
- Add 5.
+ 5
.Check:
f -1(f (x)) = f -1(3(x - 5)) =+ 5 = (x - 5) + 5 = x .
f (f -1(x)) = f (
Example 2: Find the inverse of
f (x) = 
,
x≥2
(we
must restrict the domain because
f (x)
is undefined for
x < 2
).
Original function:
- Subract 2.
- Take the square root.
- Square.
- Add 2.
Check:
f -1(f (x)) = f -1(
f (f -1(x)) = f (x 2 +2) ==
= x .
When we take the inverse of a function, the domain and range switch. In example 2, the domain of f is x≥2 and the range of f is f (x)≥ 0 . Thus, the domain of f -1 is x≥ 0 and the range of f -1 is f -1(x)≥2 .
