composite functions inverse functions piecewise functions
TRANSCRIPT
2.4
COMPOSITE FUNCTIONS INVERSE FUNCTIONS
PIECEWISE FUNCTIONS
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Composition of Functions
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
• 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))
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
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
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
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:
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
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
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
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
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
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