This document discusses function operations such as addition, subtraction, multiplication, and division. It provides examples of writing expressions for the sum, difference, product, and quotient of two functions. It also addresses the domain and range of combined functions. Specifically, it gives an example of writing functions to model the distance traveled and effect of wind for an airplane traveling at a constant speed.
2. 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.
3. You can relate speed, x, to the total stopping distance s, as a sum of R and B. Functions can also be represented by subtraction, multiplication and division.
7. 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) Solutions: P(x) = I(x) – C(x) = (250x) – (50 + 5x) = 255x – 50; P(50) = 255(50) – 50 = 12,700
8. 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.html Click on “Domain and Range” at the upper left for a tutorial.
9. 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
10. 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
11. 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) = 2x 2 + 8x Remember to Distribute!
12. 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) =
13. 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 ) = 265 x b. Write a function g ( x ) for the effect of the wind on the airplane for x hours. Solution: g ( x ) = 35 x c. Write an expression for the total speed of the airplane flying with the wind. Solution: f ( x ) + g ( x ) = 265 x + 35 x = 300 x
