1. Section 2.4 Composition and Inverse Functions 2.

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

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


17Page 80


( ( )) (2 1)f g x f x

18


2

( ( )) (2 1)

(2 1)

f g x f x

x

19Page 80


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


2( ( )) ( )g f x g x

22


2

2

( ( )) ( )

2( ) 1

g f x g x

x

23Page 80


2

2

2

( ( )) ( )

2( ) 1

2 1

g f x g x

x

x

24Page 80


2

2

( ( )) (2 1)

(2 1)

4 4 1

f g x f x

x

x x

25Page 80


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.


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


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



(a) What does f(4) represent?

32Page 80



(a) What does f(4) represent?

Bird population in the year 2007 + 4 = 2011.

33Page 80



(b) What does f −1(4) represent?

34Page 80




t = g(P) = f −1(P)

35Page 80




t = g(P) = f −1(P)

Population = input, time = output36Page 80




t = g(P) = f −1(P) → t = g(4) = f −1(4)

Population = input, time = output37Page 80




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):


Solve for R...

1( ) 40

4T f R R

40Page 81


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


The input values of the inverse function f−1 are the output values of

the function f.

43Page 81


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.


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

53

1. Section 2.4 Composition and Inverse Functions 2.

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