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Section 4.1
Inverse Functions
What are Inverse Operations?
Inverse operations are operations
that “undo” each other.
Examples
Addition and Subtraction are inverse operations
Multiplication and Division are inverse operations
How to Find Inverse Relation
• Interchange coordinates of
each ordered pair in the
relation.
• If a relation is defined by an
equation, interchange the
variables.
Inverses
• When we go from an output of a function
back to its input or inputs, we get an inverse
relation. When that relation is a function, we
have an inverse function.
• Interchanging the first and second
coordinates of each ordered pair in a
relation produces the inverse relation.
Example: Consider the relation g given by
g = {(2, 4), (3, 4), (8, 5).
Solution: The inverse of the relation is
Inverse Relation
Inverse Relation
• If a relation is defined by an equation,
interchanging the variables produces an equation of
the inverse relation.
Example: Find an equation for the inverse of the
relation:
y = x2  2x.
Solution: We interchange x and y and obtain an
equation of the
inverse:
• Graphs of a relation and its inverse are always
reflections of each other across the line y = x.
Inverse Relation
One-to-One Functions
4
5
4
3
6
3
2
7
2
1
8
1
f
f is one-to-one
6
8
9
g
g is not one-to-one
One-to-One Function
A function f is a one-to-one function if,
for two elements a and b from the domain
of f,
a  b implies f(a)

f(b)
** x-values do not share the same y-value **
Horizontal Line Test
• Use to determine whether a function is
one-to-one
• A function is one-to-one if and only if
no horizontal line intersects its graph
more than once.
Horizontal-Line Test
Graph f(x) = 3x + 4.
f(x) = 3x + 4
Example: From the
graph at the left,
determine whether the
function is one-to-one
and thus has an inverse
that is a function.
Solution: No horizontal
line intersects the
graph more than once,
so the function is oneto-one. It has an
inverse that is a
function.
Horizontal-Line Test
Graph f(x) = x2  2.
Example: From the
graph at the left,
determine whether the
function is one-to-one
and thus has an inverse
that is a function.
Solution: There are
many horizontal lines
that intersect the
graph more than once.
The inputs 1 and 1
have the same output,
1. Thus the function is
not one-to-one. The
inverse is not a
function.
Why are one-to-one
functions important?
One-to-One Functions
have
Inverse functions
Inverses of Functions
If the inverse of a function f is also a
function, it is named f 1 and read “finverse.”
The negative 1 in f 1 is not an
exponent. This does not mean the
reciprocal of f.
f 1(x) is not equal to
1
f ( x)
One-to-One Functions
A function f is
one-to-one if
different inputs have
different outputs.
That is,
if a  b
then f(a)  f(b).
A function f is
one-to-one if when
the outputs are the
same, the inputs
are the same.
That is,
if f(a) = f(b)
then a = b.
Properties of One-to-One
Functions and Inverses
• If a function is one-to-one, then its
inverse is a function.
• The domain of a one-to-one function f
is the range of the inverse f 1.
• The range of a one-to-one function f is
the domain of the inverse f 1.
• A function that is increasing over its
domain or is decreasing over its domain
is a one-to-one function.
Inverse Functions
A
f
x
B
Y = f(x)
f-1
Domain of f-1 = range of f
Range of f-1 = domain of f
How to find the Inverse of a
One-to-One Function
1. Replace f(x) with y in the equation.
2. Interchange x and y in the equation.
3. Solve this equation for y.
4. Replace y with f-1(x).
Any restrictions on x or y should be considered.
Remember: Domain and Range are interchanged
for inverses.
Example
Determine whether the function f(x) = 3x  2 is one-to-one,
and if it is, find a formula for f 1(x).
Solution:
The graph is that of a line and passes the horizontal-line test.
Thus it is one-to-one and its inverse is a function.
1. Replace f(x) with y:
2. Interchange x and y:
3. Solve for y:
4. Replace y with f 1(x):
Graph of Inverse f-1 function
• The graph of f-1 is obtained by
reflecting the graph of f across the
line y = x.
• To graph the inverse f-1 function:
Interchange the points on the graph of
f to obtain the points on the graph of
f-1.
If (a,b) lies on f, then (b,a) lies on f-1.
Example
Graph f(x) = 3x  2 and
f
using the same set of axes.
Then compare the two graphs.
1(x)
x2
=
3
Solution
x
f(x) = 3x  2
1
5
0
2
2
4
3
7
x
x+2
f 1(x) =
3
5
1
2
0
1
1
4
2
Inverse Functions and Composition
If a function f is one-to-one, then f 1 is the
unique function such that each of the following
holds:
for each x in the
1
1
( f f )( x)  f ( f ( x))  x domain of f, and
1
1
( f f )( x)  f ( f ( x))  x
for each x in the
domain of f 1.
Example
Given that f(x) = 7x  2, use composition of
functions to show that f 1(x) = (x + 2)/7.
Solution:
(f
1
1
f )( x)  f ( f ( x))
 f 1 (7 x  2)
(7 x  2)  2

7
7x

7
x
( f f 1 )( x)  f ( f 1 ( x))
x2
 f(
)
7
x2
 7(
)2
7
 x22
x
Restricting a Domain
• When the inverse of a function is not a
function, the domain of the function
can be restricted to allow the inverse
to be a function.
• In such cases, it is convenient to
consider “part” of the function by
restricting the domain of f(x). If the
domain is restricted, then its inverse is
a function.
Restricting the Domain
Recall that if a function is not one-to-one,
then its inverse will not be a function.
Restricting the Domain
If we restrict the domain values of f(x) to those greater than
or equal to zero, we see that f(x) is now one-to-one and its
inverse is now a function.