the domain of is the intersection of the domains of f and g, while the domain of f /g is the...
DESCRIPTION
Example 1 Using Operations on FunctionsTRANSCRIPT
• The domain of is the intersection of the domains of f and g, while the domain of f /g is the intersection of the domains of f and g for which
Combination on Functions
Given two functions f and g, then for all values of x for which both and are defined, the functions are defined as follows.
)(xf)(xg gfgfgfgf / and , , ,
0)(,)()()(
)()())(()()())(()()())((
xgxgxfx
gf
xgxfxgfxgxfxgfxgxfxgfSum
DifferenceProduct
Quotient
gfgfgf and , ,
.0)( xg
Example 1 Using Operations on Functions
Example 2 Using Operations on Functions
Algebraic Solutions
(a)
(b)
(c)
(d)
)0((d))5)(((c))3)(((b))1)(((a)
following. theofeach Find .53)( and 1)(Let 2
gfgfgfgf
xxgxxf
1082)1()1()1)(( gfgf
14)4(10)3()3()3)(( gfgf
5202026)5()5()5)(( gfgf
51
)0()0()0(
gf
gf
Solution (a)
(b)
(c)
).( ofdomain The)()((b)))(((a)
following. theofeach Find .12)( and 98)(Let
xgfcx
gfxgf
xxgxxf
1298)()())(( xxxgxfxgf
1298
)()()(
x
xxgxfx
gf
Domain of : Domain of : 2 1 0, solving this inequality,
1we get the interval ,2
1 1Since 0, we exclude from the domain2 2
1of . Thus the domain is , .2
f xg x
x
g
f x xg
Example 3 Using Operations on Functions
Example 4 Using Operations on Functions
Finding and Analyzing Cost, Revenue, and Profit Suppose that a businessman invests $1500 as his fixed cost in a new venture that produces and sells a device that makes programming a iPhone easier. Each device costs $100 to manufacture.
(a) Write a linear cost function with x equal to the quantity produced.(b) Find the revenue function if each device sells for $125.(c) Give the profit function for the item.(d) How many items must be sold before the company makes a profit?(e) Support the result with a graphing calculator.
Example 4 Continued
Solution(a) Using the slope-intercept form of a line, let
(b) Revenue is price quantity, so
(c) Profit = Revenue – Cost
(d) Profit must be greater than zero
or ,bmxC(x)
.1500100)( xxC
.125)( xpxxR
)()()( xCxRxP
.150025)( tosimplifies )1500100(125 xxPxx
profit. a make tosold bemust devices 61least At .60get wefor solving and ,0150025 xxx
Given find (a) and (b)
Solution(a)
(b)
,1
4)( and 12)(
x
xgxxf )2)(( gf ).3)(( fg
)]2([)2)(( gfgf 414
124)2(
g
)4(f 7181)4(2)4( f
7
)]3([)3)(( fgfg 7161)3(2)3( f
)7(g21
84
174)7(
g
21
Example 5 Composition of Functions
Let and Find (a) and (b)
Solution(a)
(b)
Note:
14)( xxf .52)( 2 xxxg ))(( xfg ).)(( xgf
)14()]([))(( xgxfgxfg
12081)52(4
)52(
2
2
2
xxxxxxf
)14(5)14(2 2 xx520)1816(2 2 xxx
52021632 2 xxx73632 2 xx
)]([))(( xgfxgf
is not always equal to . When they are, it is a special case.g f f g
Example 6 Composition of Functions
• Suppose an oil well off the California coast is leaking. – Leak spreads in circular layer over water– Area of the circle is
• At any time t, in minutes, the radius increases 5 feet every minute.– Radius of the circular oil slick is
• Express the area as a function of time using substitution.
2)( rrA
ttr 5)(
2
2
25)5(]5[)]([
tttAtrA
Example 7 Composition of Functions
Example 8 Composition of Functions
The surface area of a sphere S with radius r is S = 4 r2.
(a) Find S(r) that describes the surface area gained when r increases by 2 inches.
(b) Determine the amount of extra material needed to manufacture a ball of radius 22 inches as compared to a ball of radius 20 inches.
22 4)2(4)( rrrS
inches square extra 1056 336
204)220(4)20( 22
S
Example 10
Example 11