[email protected] MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics§9.2a§9.2a

Composite FcnsComposite Fcns



Review §Review §

Any QUESTIONS About• §9.1 → The NATURAL Base, e

Any QUESTIONS About HomeWork• §9.1 → HW-43

9.1 MTH 55



Composite FunctionsComposite Functions

In the real world, functions frequently occur in which some quantity depends on a variable that, in turn, depends on another variable.

Functions such as these are called COMPOSITE FUNCTIONS



Composing a FunctionComposing a Function

Composition with sets A & B by fcns g & f

1

3

7

A

( ) 3 1g x x

4

10

22

B

1( ) 3

2f x x

−1

2

8

C

fg

h h(x) = ?




From The Diagram notice that since f takes the output from g we can combine f and g to get a function h:

1

3

7

A

( ) 3 1g x x

4

10

22

B

1( ) 3

2f x x

- 1

2

8

C

h h(x) = ?

f (g (x)) = f (3x + 1)

1(3 1) 3

2x

3 5

2 2x

This Yields an eqn for h:3 5

( ) .2 2

h x x




The function h is the composition of f and g and is denoted f○g (read “the composition of f and g,” “f composed with g,” or “f circle g”).

1

3

7

A

( ) 3 1g x x

4

10

22

B

1( ) 3

2f x x

- 1

2

8

C

h h(x) = ?



COMPOSITION OF FUNCTIONSCOMPOSITION OF FUNCTIONS

If f and g are two functions, the composition of function f with function g is written as f○g and is defined by the equation

f og x f g x , The function where the domain of f○g

consists of those values x in the domain of g for which g(x) is in the domain of f




Graphically the f○g Domain Chain




Conceptually the f○g Operation Chain



Example Example Evaluate Composites Evaluate Composites

Given:

Find Each of the Following

a. f og 1 b. g o f 1 c. f o f 1 d. g og 1

Solution a.



Example Example Evaluate Composites Evaluate Composites

Solution b.

Solution c.

Solution d.



Example Example Fcn Composition Fcn Composition

Given f(x) = 4x and g(x) = x2 + 2, find

SOLUTION

( ) ( ( ))f g x f g x = f (x2 + 2)

= 4(x2 + 2) = 4x2 + 8




Given f(x) = 4x and g(x) = x2 + 2, find

SOLUTION ( ) ( ( ))g f x g f x = g(4x)

= (4x)2 + 2

= 16x2 + 2 This example shows

that in general ( ) ( ).f g x g f x




Given:

Find Each Composite Function

a. f og x b. g o f x c. f o f x

Solution a.




Given:

Solution b.




Given:

Solution c.



Example Example Composite Domain Composite Domain

Given:

c. Find f og x and its domain.

d. Find g o f x and its domain.

b. Find g o f 1 .a. Find f og 1 .

Solution a.




Given:

Solution b.

Soln c.

• Domain: (−∞, 0)U(0, ∞) or {x|x ≠ 0}




Given:

Soln d.• Domain: (−∞, −1)U(−1, ∞) or {x|x ≠ −1}



DEcomposing a FunctionDEcomposing a Function

Given:

Show that each of the following provides a DEcomposition of H(x)a. Express H x as f g x ,

where f x 1

x and g x 2x2 1.

b. Express H x as f g x , where f x 1

x and g x 2x2 1.



Decomposing a FunctionDecomposing a Function

Solution: a. Express H x as f g x , where f x 1

x and g x 2x2 1.



Decomposing a FunctionDecomposing a Function

Solution: b. Express H x as f g x , where f x 1

x and g x 2x2 1.



Example Example Automobile Sales Automobile Sales

A car dealer offers an 8% discount off the manufacturer’s suggested retail price (MSRP) of x dollars for any new car on his lot. At the same time, the manufacturer offers a $4000 rebate on the purchase.

a. Write a function f(x) that represents the price after the rebate.

b. Write a function g(x) that represents the price after the dealer’s discount.




c. Write the Functions (f○g)(x) & (g○f)(x). What do these Functions Represent?

d. Calculate (g○f)(x) − (f○g)(x). Interpret this odd-looking expression

Solution a.

The MSRP is x dollars, rebate is $4k, so

f(x) = x – 4000

represents the price of the car after the rebate.




Solution b.The dealer’s discount is 8% of x, or 0.08x,

so: g(x) = x – 0.08x = 0.92x represents the price of the car after the

dealer’s discount. Soln c.

• This represents the price when the DEALER’S discount is is applied first.




Solution c. (cont.)

• This represents the price when the MANUFACTURER’S rebate is applied first.




Solution d.

• This equation shows that it will cost $320 MORE for any car, regardless of its price, if you apply the rebate first and then the discount second.



WhiteBoard WorkWhiteBoard Work

Problems From §9.2 Exercise Set• 10, 12, 56, 58, 70

Composition of Functions Corresponds to a Production Line



All Done for TodayAll Done for Today

FunctionMachines

& CoDomain



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

Documents

[email protected] MTH55_Lec-58_sec_9-2a_Composite_Fcns.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical