6.4 Inverse Functions
Part 1
Goal: Find inverses of linear
functions.
Inverse Functions
A function and its inverse can be described as
the "DO" and the "UNDO" functions.
A function takes a starting value, performs some
operation on this value, and creates an output
answer.
The inverse of this function takes the output
answer, performs some operation on it, and
arrives back at the original function's starting
value.
Definition of an Inverse
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.
Notation: If f (x) is a given function, then
f -1(x) denotes the inverse of f.
Finding the Inverse of a Function
Basically, the process of finding an inverse
is simply the swapping of the x and y
coordinates.
This newly formed inverse will be a
relation, but may not necessarily be a
function.
Horizontal Line Test
The inverse of a function f is also a function
if and only if no horizontal line intersects the
graph of f more that once.
Three Methods to find the Inverse:
1. Reflect graph over the line y = x.
2. Swapping x and y-values
3. Solving Algebraically:
a. Set the function = y
b. Swap the x and y variables
c. Solve for y
Reflect the graph over y = x
Graph original function
f(x) = 2x + 3
It is drawn in blue.
If reflected over the identity
line, y = x, the original
function becomes the red
dotted graph.
Swapping x and y-values
Given relation, find the inverse relation.
x
-2
-1
0
1
2
y
4
2
0
-2
-4
x
4
2
0
-2
-4
y
-2
-1
0
1
2
Solving Algebraically
Find the equation of the inverse of the
relation f(x) = 2x – 4.
1.
Set the function = y
y = 2x – 4
2.
Swap the x and y variables
x = 2y -4
3.
Solve for y
y = (x + 4)/2
Find the equation for the Inverse
function.
f(x) = -2x + 5
f-1(x) = -(x – 5)/2
Verifying Inverse Functions
Verify that f(x) = 2x – 4 and f-1(x) = ½x + 2
are inverses.
6.3 Composition
Functions
(Inverse Functions)
Part 2
Goal: Evaluate composition
functions and prove functions are
inverses of each other using
composition functions.
Composition of Functions
The composition of a function g with a
function f is:
h(x) = g(f(x))
Example 1
Let f(x) = 3x – 14 and g(x) = x + 5. What is
the value of g(f(4))?
•First find f(4).
f(x) = 3x – 14
f(4) = 3(4) – 14 = -2
•Then find g(-2).
g(x) = x + 5
g(-2) = -2 + 5 = 3
Example 2
Let f(x) = 3x + 2 and g(x) = 2x – 7. Find
f(g(x)).
•Find g(x) first.
g(x) = 2x – 7
•Then find f(2x – 7)
f(x) = 3x + 2
f(2x – 7) = 3(2x – 7) + 2
f(2x – 7) = 6x – 21 + 2
f(2x – 7) = 6x – 19
Therefore, f(g(x)) = 6x – 19
This "DO" and "UNDO" process can be
stated as a composition of functions.
If functions f and g are inverse functions,
f(g(x)) = g(f(x)) = x.

Example: If f(x) = x-1 and g(x) = x +1
then f(g(x)) = x and g(f(x)) = x
Think of them as "undoing" one another
and leaving you right where you started.
Verify that f and g are inverse
functions.
f(x) = 3x – 1 and g(x) = ⅓x + ⅓