composite functions inverse functions piecewise functions

2.4

COMPOSITE FUNCTIONS INVERSE FUNCTIONS

PIECEWISE FUNCTIONS

Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

Composition of Functions


• For two functions f(t) and g(t), the function f ( g(t)) is said to be a composition of f with g.

• The function f(g(t)) is defined by using the output of the function g as the input to f.

))(())(( xgfxgfalso

Composition of Functions Example 3 (a)

Let f(x) = 2x + 1 and g(x) = x2 − 3.

(a) Calculate

f(g(3))

and

g(f(3))


Composition of Functions Solution (a)

g(3) = (3)2 − 3 = 6, so f(g(3)) = f(6)

f(6) = 2(6) + 1 = 13, so f(g(3)) = 13


To calculate g(f(3)), we have

f(3) = 2(3)+1=7 g(f(3)) = g(7)

g(7) = (7)2 − 3 = 46, so g(f(3)) = 46

Note in this case, f(g(3)) ≠ g(f(3)).

Composition of Functions Solution (b)

In general, the functions f(g(x)) and g(f(x)) are different:


f(g(x)) = f(x2 – 3) = 2(x2 – 3) + 1 = 2x2 – 6 + 1 = 2x2 – 5

g(f(x)) = g(2x + 1) = (2x + 1)2 – 3 = 4x2 + 4x + 1 – 3 = 4x2 + 4x – 2

A circular oil slick is expanding with radius, r in yards, at time t in hours given by , for t in hours, 0< t < 10. Find a formula for the area in square yards, A = f(t), as a function of time.

21.02 ttr

substitute

then simplify

21.02 ttr

2rA

22 )1.02( ttA

)01.04.04( 422 tttA

foilttttA )1.02)(1.02( 22

)01.02.02.04( 4222 ttttA

Give the meaning and units of the composite function

R(f (p)), where Q = f (p) is the number of barrels of oil sold by a company when the price is p dollars/barrel and R(Q) is the revenue earned in millions of dollars.

R(f (p)) price /barrel

# barrels of oil

revenue earned

so, revenue f (price)

))(()),((

)),1(()),0((

:

32)(

,1)(: 2

xfgxgf

gfgf

find

xxg

xxfgiven

10 26

101241)32( 22 xxx 5423)1(2 22 xxx

The roles of a function’s input and output can sometimes be reversed.

• 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

Inverse Function Procedure

Reassign the variables, then solve for y


1fNOT AN EXPONENT

?)0( f

0(?)1 f

0(?) f

?)0(1 f2

3

2

3

INVERSES

(10.2)Generate inverse Omit:17-20, 25, 26.

#15

Inverse maybe a function yx

yx

yx

xy

3

3

3

3

12

12

12

variablesreassign

12

1f

#16

Inverse maybe a function

yx

x

xyx

xyyx

yxxy

yyx

y

yx

reassign

x

xy

21

)21(

2

2

)12(

12

12

1f

#23

yyx

y

yx

y

yx

reassign

x

xy

74)4(

4

74

4

74

4

74

2

2

#23 continued

yx

x

xyx

yxyx

yyxx

2

2

22

22

22

7

44

)7(44

744

744

1f

#24 Generate the inverse

x

xy

11

3

Generate the inverse

45

5

38

72

x

xy

Generate the inverse

x

xy9

2

45

Figure 10.18 defines the function f. Rank the following quantities in order from least to greatest:

)3(),3(,3),0(),0(,0 11 ffff

3)0()0(0)3()3( 11 ffff

#81 Use Figure 10.34 (a) Evaluate f (g (a)).

(b) Evaluate g ( f (c)).

(c) Evaluate f (b) − g (b).

(d) For what positive value(s) of x is

f (x) ≤ g (x)?

(a) Evaluate f (g (a))

a

(b) Evaluate g ( f (c))

b

(c) Evaluate )()( 11 bgbf

cc )(0

(d) For what positive value(s) of x is

f (x) ≤ g (x)?

ax

HW: A company believes there is a linear relationship between the consumer demand for its products and the price charged. When the price was $3 per unit, the quantity demanded was 500 units per week. When the unit price was raised to $4, the quantity demanded dropped to 300 units per week. Let D(p) be the quantity per week demanded by consumers at a unit price of $p.

(a) Estimate and interpret D(5).

(b) Find a formula for D(p) in terms of p.

(c) Calculate and interpret D-1(5).

(d) Give an interpretation of the slope of D(p) in terms of demand.

(e) Currently, the company can produce 400 units every week. What should the price of the product be if the company wants to sell all 400 units?

(f) If the company produced 500 units per week instead of 400 units per week, would its weekly revenues increase, and if so, by how much?

The predicted pulse in beats per minute (bpm) of a healthy person fifteen minutes after consuming q milligrams of caffeine is given by r = f (q). The amount of caffeine in a serving of coffee is qc and rc = f(qc ). Assume that f is an increasing function for non-toxic levels of caffeine. What do each of the following statements tell you about caffeine and a person's pulse?

))(1.1()6

)20()5

)0()()4

20)(2)3

)20()2

)2()1

c

1

cc

1

c

c

1

c

1

c

qff

qrf

fqf

rf

rf

qf

2.3

PIECEWISE DEFINED FUNCTIONS


KNOW

BASIC GRAPHS

Without a calculator, sketch all four functions on the same axis and label

each along with the coordinates of all intercepts and intersecting points

( ) 4 ( ) 5

( ) 2 4 ( ) 5

f x x g x x

h x x j x x

composite functions inverse functions piecewise functions

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