1
Powerpoint slides copied from or based upon:
Connally,
Hughes-Hallett,
Gleason, Et Al.
Copyright 2007 John Wiley & Sons, Inc.
Functions Modeling Change
A Preparation for Calculus
Third Edition
Section 2.4
Composition and Inverse Functions
2
Composition and Inverse Functions
Two functions may be connected by the fact that the output of one is the
input of the other.
3Page 79
Let's define a new function:
Cost (C) as a function of # of gallons of paint (n):
( ) $30.50C g n n
4Page 79
Cost (C) as a function of # of gallons of paint (n):
Previously, we saw- # of gallons of paint (n) as a function of house Area (A):
( ) $30.50C g n n
( )250
An f A
5Page 79 Example #1
Now we want Cost (C) as a function of house Area (A):
( ) $30.50
( )250
C g n n
An f A
6Page 79
Now we want Cost (C) as a function of house Area (A):
( ) $30.50
( )250
C g n n
An f A
( ) & ( )
( ) ( ( ))
C g n n f A
C h A g f A
7Page 79
( ) $30.50
( )250
( ) & ( )
( ) ( ( ))
30.50 0.122250
C g n n
An f A
C g n n f A
C h A g f A
AA
8Page 79
( ) $30.50
( )250
( ) & ( )
( ) ( ( ))
30.50 0.122250
C g n n
An f A
C g n n f A
C h A g f A
AA
h = composition of functions f & g
f = inside function, g = outside function 9Page 79
You will recall (!) Temperature (T) as a function of chirp Rate (R):
140
4T R
10Page 79 Example 2
You will recall (!) Temperature (T) as a function of chirp Rate (R):
Let's define a new function- Chirp Rate (R) as a function of time (x):
Here, x is in hrs. since midnight & 0 ≤ x ≤ 10
2( ) 20R g x x
140
4T R
11Page 79
Now we want Temperature (T) as a function of time (x):
2
140
4
( ) 20
T R
R g x x
12Page 79
Now we want Temperature (T) as a function of time (x):
2
140
4
( ) 20
( ) & ( )
( ) ( ( ))
T R
R g x x
T f R R g x
T h x f g x
13Page 79
2
22
140
4
( ) 20
( ) & ( )
( ) ( ( ))
1(20 ) 40 45
4 4 (0 10)
T R
R g x x
T f R R g x
T h x f g x
xx
x
14Page 79
2
22
140
4
( ) 20
( ) & ( )
( ) ( ( ))
1(20 ) 40 45
4 4 (0 10)
T R
R g x x
T f R R g x
T h x f g x
xx
x
h = composition of functions f & g
f = outside function, g = inside function
15Page 79
If f(x) = x2 and g(x) = 2x + 1, find (a) f(g(x))
(b) g(f(x))
16Page 80 Example #3
If f(x) = x2 and g(x) = 2x + 1, find (a) f(g(x))
17Page 80
If f(x) = x2 and g(x) = 2x + 1, find (a) f(g(x))
( ( )) (2 1)f g x f x
18
If f(x) = x2 and g(x) = 2x + 1, find (a) f(g(x))
2
( ( )) (2 1)
(2 1)
f g x f x
x
19Page 80
If f(x) = x2 and g(x) = 2x + 1, find (a) f(g(x))
2
2
( ( )) (2 1)
(2 1)
4 4 1
f g x f x
x
x x
20Page 80
If f(x) = x2 and g(x) = 2x + 1, find (b) g(f(x))
21Page 80
If f(x) = x2 and g(x) = 2x + 1, find (b) g(f(x))
2( ( )) ( )g f x g x
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If f(x) = x2 and g(x) = 2x + 1, find (b) g(f(x))
2
2
( ( )) ( )
2( ) 1
g f x g x
x
23Page 80
If f(x) = x2 and g(x) = 2x + 1, find (b) g(f(x))
2
2
2
( ( )) ( )
2( ) 1
2 1
g f x g x
x
x
24Page 80
If f(x) = x2 and g(x) = 2x + 1, find (a) f(g(x))
2
2
( ( )) (2 1)
(2 1)
4 4 1
f g x f x
x
x x
25Page 80
If f(x) = x2 and g(x) = 2x + 1, find (b) g(f(x))
2
2
2
( ( )) ( )
2( ) 1
2 1
g f x g x
x
x
26Page 80
Inverse Functions
27Page 80
Inverse Functions
The roles of a function's input and output can sometimes be
reversed.
28Page 80
Inverse Functions
Example: the population, P, of birds is given, in thousands, by P = f(t), where t is the number of years since 2007. (Here t = input, P = output.) Define a new function, t = g(P), which tells us the value of t given the value of P instead of the other way round. (Here, P = input, t = output.) The functions f and g are called inverses of each other. A function which has an inverse is said to be invertible.
29Page 80 Example #4
Inverse Function Notation
f-inverse: f−1 (not an exponent!)
Back to our example:
P = f(t) original functiont = g(P) = f −1(P) inverse function
30Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(a) What does f(4) represent?(b) What does f −1(4) represent?
31Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(a) What does f(4) represent?
32Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(a) What does f(4) represent?
Bird population in the year 2007 + 4 = 2011.
33Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(b) What does f −1(4) represent?
34Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(b) What does f −1(4) represent?
t = g(P) = f −1(P)
35Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(b) What does f −1(4) represent?
t = g(P) = f −1(P)
Population = input, time = output36Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(b) What does f −1(4) represent?
t = g(P) = f −1(P) → t = g(4) = f −1(4)
Population = input, time = output37Page 80
Back to our example:
Using P = f(t), (P = bird pop. in thousands, t = number of yrs. since 2007):
(b) What does f −1(4) represent?
t = g(P) = f −1(P) → t = g(4) = f −1(4)
f −1(4) = # of years (since 2007) at which there were 4,000 birds on the island.
38Page 80
You will recall (!!) Temperature (T) as a function of chirp Rate (R):
What is the formula for the inverse function, R= f −1(T)?
1( ) 40
4T f R R
39Page 81 Example 5
You will recall (!!) Temperature (T) as a function of chirp Rate (R):
What is the formula for the inverse function, R= f −1(T)?
Solve for R...
1( ) 40
4T f R R
40Page 81
What is the formula for the inverse function, R= f −1(T)?
Solve for R...
1
140
41
404
4( 40)
4 160 ( )
T R
T R
T R
T R f T
41Page 81
Domain & Range of an Inverse Function
42Page 81
Domain & Range of an Inverse Function
The input values of the inverse function f−1 are the output values of
the function f.
43Page 81
Domain & Range of an Inverse Function
The input values of the inverse function f−1 are the output values of
the function f.
Therefore, the domain of f−1 is the range of f.
44Page 81
What about the domain & range of the cricket function T=f(R) and the
inverse R= f −1(T)?
1
1( ) 40
4
( ) 4 160
T f R R
R f T T
45Page 81
1
1( ) 40
4
( ) 4 160
T f R R
R f T T
46Page 81
1
1( ) 40
4
( ) 4 160
T f R R
R f T T
1( ) 40
4T f R R For
if a realistic domain is 0 ≤ R ≤ 160, then the range of f is 40 ≤ T ≤ 80.
47Page 81
A Function and its Inverse Undo Each Other
48Page 81
A Function and its Inverse Undo Each Other
Calculate the composite functions:
f−1(f(R)) & f(f−1(T))
for the cricket example.
Interpret the results.
49Page 81 Example #6
1
1
1( ) 40
4
( ) 4 160
1( ( )) 4 40 160
4
160 160
T f R R
R f T T
f f R R
R
R
50Page 81
1
1
1( ) 40
4
( ) 4 160
1( ( )) 4 160 40
440 40
T f R R
R f T T
f f T T
T
T
51Page 81
The functions f and f−1 are called inverses because they “undo” each
other when composed.
52Page 81
End of Section 2.4
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