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Function Operations
We all know that the faster you drive the farther you will travel while reacting to an obstacle and while braking to complete that stop. The reaction distance and braking distance depend on the speed of the vehicle at the moment the obstacle is observed.
The functions below represent speed, x, in miles per hour to the reaction distance, R(x), and the braking distance B(x), both in feet.
2
191
)(
1011
)(
xxB
xxR
You can relate speed, x, to the total stopping distance s, as a sum of R and B.
2
191
)(
1011
)(
xxB
xxR
2
191
1011
)(
)()()(
xxxS
xBxRxS
Functions can also be represented by subtraction, multiplication and division.
Operations with Functions
0)()()(
))((
)()())((
)()())((
)()())((
xgxgxf
xgf
xgxfxgf
xgxfxgf
xgxfxgf
.5)(4)( 2 xxgandxxfLet
Let’s try some examples:
Write an expression for each function. State any restrictions.
))((.
))((.
))((.
))((.
xgfD
xfg
C
xfgB
xgfA
2054
)(
)5)(4(
))((.
!
!445
)4()5(
))((.
1
,
)4()5(
))((.
9
)5()4(
))((.
23
2
2
2
2
2
2
2
xxx
FOILDistribute
xx
xgfD
furtheranySimplifiedbeCannot
zerobydivideCannotxxx
xx
xfg
C
xx
TermsLikeAddthenSignNegativetheDistribute
xx
xfgB
TermsLikeAdd
xx
xx
xgfA
Solutions:
More examples:
A craftsmen makes and sells antique chests.
The function represents his cost in dollars to produce x chest.
The function represents the income in dollars form selling x chest.
Write and simplify a function P(x) that represents the profit from selling x chest.Find P(50), the profit earned when he makes and sells 50 chest.(Remember: Profit = Income – Cost)
xxC 550)(
xxI 250)(
Solutions:P(x) = I(x) – C(x) = (250x) – (50 + 5x) = 255x – 50; P(50) = 255(50) – 50 = 12,700
Domain and Range
The domain of the sum, difference, product and quotient functions consist of the x-values such that are in the domains of BOTH f and g.
However, the domain of a quotient function does not contain any x-value for which g(x) = 0.Follow this link to review Domain and Range:http://www.algebasics.com/3way12.htmlClick on “Domain and Range” at the upper left for a tutorial.
The domain for f(x) is all real numbers because there are no values of x such that the function will not be defined.
The domain for g(x) is all real numbers.
Therefore, the domain for (f + g)(x) is all real numbers.
Examples:
F(x) = x + 4 and G(x) = 2x
(f + g)(x) = (x + 4) + (2x) = 3x + 4
The domain for f(x) is all real numbers because there are no values of x such that the function will not be defined.
The domain for g(x) is all real numbers.
Therefore, the domain for (f - g)(x) is all real numbers.
Examples:
f(x) = x + 4 and g(x) = 2x
(f - g)(x) = (x + 4) - (2x) = -x + 4
The domain for f(x) is all real numbers because there are no values of x such that the function will not be defined.
The domain for g(x) is all real numbers.
Therefore, the domain for (f • g)(x) is all real numbers.
Examples:
f(x) = x + 4 and g(x) = 2x
(f • g)(x) = (x + 4)(2x) = 2x2 + 8x
Remember to Distribute!
The domain for f(x) is all real numbers because there are no values of x such that the function will not be defined.
The domain for g(x) is all real numbers.
However, the domain of a quotient function does not contain any x-value for which g(x) = 0.
If the value for x in 2x cannot produce 0, then x cannot equal zero.
Therefore, the domain for (f ÷ g)(x)is all real numbers except 0.
Examples:
f(x) = x + 4 and g(x) = 2x
(f ÷ g)(x) = (x + 4)÷(2x) = x
x2
4
Another example:
An airplane travels at a constant speed of 265 miles per hour in still air. During a particular portion of the flight, the wind speed is 35 miles per hour in the same direction the plane is flying.
a. Write a function f(x) for the distance traveled by the airplane in still air for x hours.
Solution: f(x) = 265x
b. Write a function g(x) for the effect of the wind on the airplane for x hours.
Solution: g(x) = 35x
c. Write an expression for the total speed of the airplane flying with the wind.
Solution: f(x) + g(x) = 265x + 35x = 300x
Helpful Links:
http://www.purplemath.com/modules/fcnops.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut30b_operations.htm
