Chabot Mathematics
§9.2b
Inverse Fcns
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Review § 9.2
MTH 55
 Any QUESTIONS About
• §9.2 → Composite Functions
 Any QUESTIONS About HomeWork
• §9.2 → HW-43
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Inverse & One-to-One Functions
 Let’s view the following two functions as
relations, or correspondences:
Toys
States
Domain
(inputs)
Maine
Illinois
Iowa
Ohio
Chabot College Mathematics
3
Range
(outputs)
1
7
2
3
Domain
(inputs)
ball
rope
phone
car
Range
(outputs)
Ann
Jim
Jack
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Inverse & One-to-One Functions
 Suppose we reverse the arrows. We
obtain what is called the inverse relation.
Are these inverse relations functions?
Toys
States
Range
(inputs)
Maine
Illinois
Iowa
Ohio
Chabot College Mathematics
4
Domain
(outputs)
1
7
2
3
Range
(inputs)
ball
rope
phone
Car
Domain
(outputs)
Ann
Jim
Jack
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Inverse & One-to-One Functions
Toys
States
Range
(inputs)
Maine
Illinois
Iowa
Ohio
Domain
(outputs)
1
7
2
3
Range
(inputs)
ball
rope
phone
Car
Domain
(outputs)
Ann
Jim
Jack
 Recall that for each input, a function
provides exactly one output. The inverse
of “States” correspondence IS a function,
but the inverse of “Toys” is NOT.
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
One-to-One for “States” Fcn
 In the States function, different
inputs have different outputs, so it
is a one-to-one function.
 In the Toys function, rope and
phone are both paired with Jim.
Toys
Range
Domain
 Thus the
(inputs)
(outputs)
Toy function is
ball
Ann
rope
Jim
NOT one-to-one. phone
Jack
Car
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
One-to-One Summarized
 A function f is one-to-one if different
inputs have different outputs. That
is, if for a and b in the domain of f
with a ≠ b we have f(a) ≠ f(b)
then the function f is one-to-one.
 If a function is one-to-one, then its
INVERSE correspondence is
ALSO a FUNCTION.
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
One-to-One Fcn Graphically
 Each y-value in
the range
corresponds to
only one x-value
in the domain
• i.e.; Each x has
a Unique y
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
NOT a One-to-One Fcn
 The y-value y2 in
the range
corresponds to
TWO x-values, x2
and x3, in the
domain.
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
NOT a Function at All
 The x-value x2 in
the domain
corresponds to
the TWO
y-values, y2 and
y3, in the range.
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Definition of Inverse Function
 Let f represent a one-to-one
function. The inverse of f is also a
function, called the inverse function
of f, and is denoted by f−1.
 If (x, y) is an ordered pair of f, then
(y, x) is an ordered pair of f−1, and
we write x = f−1(y). We have
y = f (x) if and only if f−1(y) = x.
Chabot College Mathematics
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  f-values ↔ f-1-values

Assume that f is a one-to-one function.
a. If f(3) = 5, find f-1(5)
b. If f-1(−1) = 7, find f(7)

Solution: Recall that y = f(x) if and
only if f-1(y) = x
a. Let x = 3 and y = 5. Now 5 = f(3) if and
only if f−1(5) = 3. Thus, f−1(5) = 3.
b. Let y = −1 and x = 7. Now, f−1(−1) = 7 if
and only if f(7) = −1. Thus, f (7) = −1.
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Inverse Function Property
 Let f denote a one-to-one function. Then
1.
f o f x   f f x  x
1
1
for every x in the domain of f–1.
2.
f
1
of
x   f  f x  x
1
for every x in the domain of f .
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Inverse Fcn Property
 Let f(x) = x3 + 1. Show that f 1( x)  3 x  1.
 Soln:  f
f 1  ( x)  f  f 1 ( x)   f


3 x 1


3
3
 x 1 1
 x 11  x
f
1
f  ( x)  f
1
 f ( x)   f  x
1
3
 1
3
3
 ( x  1)  1
3 3
 x x
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
UNIQUE Inverse Fcn Property
 Let f denote a one-to-one function.
Then if g is any function such that
f g x   x for every x in the domain of g and
g  f x   x for every x in the domain of f, then
g = f –1. That is, g is the inverse function of f.
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Verify Inverse Functions
 Verify that the following pairs of
functions are inverses of each other:
x3
f x   2x  3 and g x  
.
2
 Solution: From the composition of f & g.
 x  3
 f og x   f g x  f 

2
 x  3
 2
 3 x  3 3

 2 
Chabot College Mathematics
16
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Verify Inverse Functions
 Solution (cont.): Now Find g  f x .
g o f x   g  f x  g 2x  3
2x  3  3


2
x
 Observe: f g x   g  f x   x,
 This Verifies that f and g are indeed
inverses of each other.
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Find Inverse of a Fcn
 Given that f(x) = 5x − 2 is one-to-one,
then find an equation for its inverse
 Solution: f (x) = 5x – 2
y = 5x – 2
Replace f(x) with y
x = 5y – 2
Interchange x and y
x2
y
5
x2
1
f ( x) 
5
Chabot College Mathematics
18
Solve for y
Replace y with f-1(x)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Procedure for finding f−1
1. Replace f(x) by y in the equation
for f(x).
2. Interchange x and y.
3. Solve the equation in Step 2 for y.
4. Replace y with f−1(x).
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Find the Inverse
 Find the inverse of the f x   x  1 , x  2.
one-to-one function
x2
 Solution: Step 1
x 1
y
x2
Step 2
y 1
x
y2
Step 3
x y  2   y  1
xy  2 x  y  1
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Find the Inverse
Step 3
(cont.)
xy
x 22x
x yy  yy  1122x
x  yy
xy 22y
xy  y  2x  1
y x  1  2x  1
2x  1
y
x 1
Step 4
Chabot College Mathematics
21
2x  1
f x  
, x 1
x 1
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Find Domain & Range
 Find the Domain &
x 1
f x  
, x  2.
Range of the function
x2
 Solution: Domain of f, all real numbers
x such that x ≠ 2, in interval notation
(−∞, 2)U(2, −∞)
 Range of f is
2x  1
1
f x  
, x 1
−1
the domain of f
x 1
 Range of f is (−∞, 1) U (1, −∞)
Chabot College Mathematics
22
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Inverse Function Machine
 Let’s consider inverses of functions in
terms of function machines. Suppose
that a one-to-one function f, has been
programmed into a machine.
 If the machine has a reverse switch,
when the switch is thrown, the machine
performs the inverse function, f−1.
Inputs then enter at the opposite end,
and the entire process is reversed.
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Reverse Switch Graphically
Reverse
Forward
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Horizontal Line Test
 Recall that to be a
Function an (x,y)
relation must pass the
VERTICAL LINE test
 In order for a function
to have an inverse
that is a function, it
must pass the
HORIZONTAL-LINE  NOT a Function –
Fails the Vertical
test as well
Line Test
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Horizontal Line Test Defined
 If it is impossible to
draw a horizontal line
that intersects a
function’s graph more
than once, then the
function is
one-to-one.
 For every one-to-one
function, an inverse
function exists.
Chabot College Mathematics
26
 A Function withOUT
and Inverse – Fails
the Horizontal Line
Test (not 1-to-1)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Horizontal Line Test
 Determine whether the function f(x) = x2 + 1 is
one-to-one and thus has an inverse fcn.
 The graph of f is shown. Many
horizontal lines cross the graph
more than once. For example,
the line y = 2 crosses where the
first coordinates are 1 and −1.
Although they have different
inputs, they have the same
output: f(−1) = 2 = f(1). The
-5
function is NOT one-to-one,
therefore NO inverse function
exists
Chabot College Mathematics
27
y
8
7
6
5
4
3
2
1
-4 -3 -2 -1
1
-1
2 3 4 5
-2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
x
Example  Horizontal Ln Test
 Use the horizontal-line test to determine
which of the following fcns are 1-to-1
2
2
f x   2x
c. hxx 1 2 c.x h x   2
a.a.f5x b.
  g2xx 5 x b. g1 xb.
 Soln a.
• No horizontal line
intersects the graph of
f in more than one
point, therefore the
function f is one-to-one
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Horizontal Ln Test
 Use the horizontal-line test to determine
which of the following fcns are 1-to-1
2
f x   2xa. 5 b. g x   x  1 b.
c. h x   2 x
 Soln b.
• No horizontal line
intersects the
graph of f in more
than one point,
therefore the
function f is 1-to-1
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Graphing Fcns and Their Inverses
 How do the
graphs of a
function and
its inverse
compare?
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Graphs Inverse Fcn
 Graph f(x) = 5x − 2 and f−1(x) = (x + 2)/5
on the same set of axes and compare
f (x) = 5x – 2
 Solution:
 Note that the graph of
f−1(x) can be drawn by
reflecting the graph of f
across the line y = x.
 When x and y are
interchanged to find a
formula for f−1(x), we are,
in effect, Reflecting or
Flipping the graph of f.
Chabot College Mathematics
31
6
5
4
3
2
1
-5 -4 -3 -2 -1
1
-1
-2
-3
-4
2 3 4 5
x
f -1(x) = (x + 2)/5
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Visualizing Inverses
 The graph of
f−1 is a
REFLECTION
of the graph of
f across the
line y = x.
Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Use y = x Mirror Ln
 The graph of the function f is shown at
Lower Right. Sketch the graph of the f−1
 Soln
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Inverse or Not?
Ray’s Music Mart has six employees. The
first table lists the first names and the Social
Security numbers of the employees, and the
second table lists the first names and the
ages of the employees

a. Find the inverse of the function defined by the
first table, and determine whether the inverse
relation is a function
b. Find the inverse of the function defined by the
second table, and determine whether the
inverse relation is a function
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Inverse or Not?
Dwayne
590-56-4932
Sophia
599-23-1746
Desmonde 264-31-4958
Carl
432-77-6602
Anna
195-37-4165
Sal
543-71-8026
Chabot College Mathematics
35
 Solution:
Every y-value
corresponds to
exactly one
x-value. Thus
the inverse of
the function
defined in this
table is a
function
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Inverse or Not?
Dwayne
24
Sophia
26
Desmonde 42
Carl
51
Anna
24
Sal
26
Chabot College Mathematics
36
 Solution:
There is more than one
x-value that corresponds
to a y-value.
For example, the age of
24 yields the names
Dwayne and Anna.
Thus the inverse of the
function defined in this
table is NOT a function.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  Hydrostatic Pressure
 The formula for finding the water
pressure p (in pounds
15d
per square inch, or psi),
p

at a depth d (in feet)
33
below the surface is
.
 A pressure gauge on a Diving Bell breaks
and shows a reading of 1800 psi.
Determine how far below the surface the
bell was when the gauge failed
Chabot College Mathematics
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Example  HydroStatic P
15d
 Solution: The depth is
p
.
given by the inverse of
33
15d
 Solve the
p
Let
p
=
1800
psi
Inverse
33
33 1800 
Eqn for p 33p  15d
d
15
33p
d
d  3960
15
 The Diving Bell was 3960 feet below the
surface when the gauge failed
Chabot College Mathematics
38
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
WhiteBoard Work
 Problems From §9.2 Exercise Set
• 38, 42, 60, 68, 76
 Some
Temperature
Scales
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
All Done for Today
Old Style
Diving
Bell
Chabot College Mathematics
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-59_Fa08_sec_9-2b_Inverse_Fcns.ppt