Inverse Functions, one to one and inverse functions

One-to-one and Inverse
Functions

Review:
A is any set of ordered pairs.
A function does not have any y values
repeated.
A is a set of ordered pairs where
x is not repeated.
Only functions can have
functions.

What is an Inverse?
Examples:
f(x) = x – 3 f-1(x) = x + 3
g(x) = , x ≥ 0 g-1(x) = x2 , x ≥ 0
h(x) = 2x h-1(x) = ½ x
k(x) = -x + 3 k-1(x)= -(x – 3)
x
An inverse relation is a relation that performs the
opposite operation on x (the domain).

Illustration of the Definition of
Inverse Functions

The ordered pairs of the function f are reversed to
produce the ordered pairs of the inverse relation.
Example: Given the function
f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4}
and its range is {1, 2, 3}.
The inverse of f is {(1, 1), (3, 2), (1, 3), (2, 4)}.
The domain of the inverse relation is the range of the
original function.
The range of the inverse relation is the domain of the
original function.

How do we know if an inverse
function exists?
• Inverse functions only exist if the original function
is one to one. Otherwise it is an inverse relation
and cannot be written as f-1(x).
• What does it mean to be one to one?
That there are no repeated y values.

x
y
2
2
Horizontal Line Test
Used to test if a function is one-to one
If the line intersection more than once then it is not one to
one.
Therefore there is not inverse function.
y = 7
Example: The function
y = x2 – 4x + 7 is not one-to-one
because a horizontal line can
intersect the graph twice.
Examples points: (0, 7) & (4, 7).
(0, 7) (4, 7)

one-to-one
The Inverse is a Function
Example: Apply the horizontal line test to the graphs
below to determine if the functions are one-to-one.
a) y = x3 b) y = x3 + 3x2 – x – 1
not one-to-one
The Inverse is a Relation
x
y
-4 4
4
8
x
y
-4 4
4
8

y = x
The graphs of a relation and its inverse are reflections
in the line y = x.
The ordered pairs of f are given by
the equation .
4
)
2
( 3


y
x
4
)
2
( 3


x
y
Example: Find the graph of the inverse relation
geometrically from the graph of f(x) = )
2
(
4
1 3

x
x
y
2
-2
-2
2
The ordered pairs of the inverse are
given by .
)
2
(
4
1 3

 x
y
)
2
(
4
1 3

 y
x

The inverse of a relation is the reflection in the line y = x of the graph.
1
: ( )
: ( )
Function f x
Inverse f x


Graph of an Inverse Function
Functions and their
inverses are symmetric
over the line y =x

NOTE: It is important to remember that not every function has an inverse.

2
( 1) 2
y x
  
The circled part of the expression is a square
so it will always be > 0. The smallest value it
can be is 0. This occurs when x = 1. The
vertex is at the point (1, −2 )
Examples:

Find the inverse:
Graph and find domain and range.
Is the inverse a function?
2
2
y x
  
functio
n f(x)
inverse
f –1 (x)
D
R

Find the inverse:
( ) 2 2
f x x
 
functio
n f(x)
inverse
f –1 (x)
D
R

Find the inverse:
What can you say is true of all cubic functions?
3
( ) 1
f x x
 
functio
n f(x)
inverse
f –1 (x)
D
R

DETERMINING IF 2 FUNCTIONS ARE INVERSES:
The inverse function “undoes” the original function,
that is, f -1( f(x)) = x.
The function is the inverse of its inverse function,
that is, f ( f -1(x)) = x.
Example: The inverse of f(x) = x3 is f -1(x) = .
x
3
f -1( f(x)) = = x and f ( f -1(x)) = ( )3 = x.
3
x
3
x
3

It follows that g = f -1.
Example: Verify that the function g(x) =
is the inverse of f(x) = 2x – 1.
f(g(x)) = 2g(x) – 1 = 2( ) – 1 = (x + 1) – 1 = x
2
1

x
2
1

x
g( f(x)) = = = = x
2
)
1
)
1
2
(( 

x
2
2x
2
)
1
)
(
( 
x
f

Review of Today’s Material
• A function must be 1-1 (pass the horizontal
line test) to have an inverse function (written
f-1(x)) otherwise the inverse is a relation (y =)
• To find an inverse: 1) Switch x and y
2) Solve for y
• Given two relations to test for inverses.
f(f-1(x)) = x and f-1(f(x)) = x **both must be true**
• Original and Inverses are symmetric over y =x
• “ “ ” have reverse domain & ranges

Inverse Functions, one to one and inverse functions

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