INVERSE
FUNCTIONS
BY: MPEO KHATALA
What is an inverse function?
The inverse of a function is the set of ordered pairs obtained by interchanging the first and
second elements of each pair in the original function.
In plain English, finding an inverse is simply the swapping of the x and y coordinates.
f (x) = {(2,3), (4,5), (-2,6), (1,-5)} (function)
The inverse of f (x) = {(3,2), (5,4), (6,-2), (-5,1)} (function)
Let's look at another example: g (x) = {(4,1), (8,3), (-5,3), (0,1)} (function)
The inverse of g (x) = {(1,4), (3,8), (3,-5), (1,0)} (NOT a function, x's repeat)
Definition: Inverse of a Function: The relation formed when the independent variable (x) is
exchanged with the dependent variable (y) in a given relation. (This inverse may NOT be a
function.)
Definition: Inverse Function: If the inverse of a function (defined above) is itself also a
function, it is then called an inverse function.
FINDING AN INVERSE
A function receives an input value, performs some operation on this value, and creates an output
answer. The inverse of the function takes that output answer, performs some operation on it, and
arrives back at the original function's starting input value.
A function and its inverse "undo" one another, leaving you right back where you started.
So how do we find these inverse relationships?
Method 1
If the function is stated as a set of ordered pairs, we can find the inverse of the function by simply
swapping the ordered pairs (as seen in the second slide).
If the definition of the function is fairly simple, we may be able to find the inverse of the function
by examining selected ordered pairs and looking for a relationship for the inverse.
Let f(x)= x + 1
Inverse of f(x)
x
y = f(x)
x
Inverse
1
2
2
1
3
-4
4
-3
-2
-1
-1
-2
0
1
1
0
Method 2 (algebraic):
There is a simple method to find the inverse of a function algebraically.
Find the inverse of function f (x) = x + 6.
1. y = x + 6
2. y - 6 = x
3. x - 6 = x which is the inverse f -1 (x) = x – 6
Process
1. Set the function = y
2. Make x the subject
3. Change your y to x
References