Tuesday, October 9, 2012

In this post, I will explain the composition of functions

Suppose we wish to represent the function g(x) by a token machine. The token machine yields one token for every quarter that is deposited into the machine. The tokens can be used to purchase prizes. We think of the quarters as the input x and the number of tokens as the output g(x). Suppose further that there is another machine that requires the use of tokens to obtain prizes. A certain number of tokens are needed to purchase each prize. We'll define the prize machine as f(x). The input is the number of tokens, which we defined as g(x). Notice that purchasing a prize out of the second machine is dependent on the number of tokens from the first machine. Such dependence can be interpreted in mathematical terms as composition of functions.

In the previous example, the domain x yields g(x), the number of tokens. Then g(x) becomes the input into f(x) to produce the output, which is the prize purchased. The end result is a composition function f º g, also noted as f(g(x)).Next, notice the composition function f º g, also noted as f(g(x)), joining the two machines together as one machine which automatically deposits tokens into the prize generator, which ejects the appropriate prize corresponding to the number of tokens generated.

Examples: Find f(g(x)), f(g(-6)) and f(g(10)) if f(x) = x^2 + 4 and g(x) = 5 – x
.
To find f(g(x)) is the same as finding f(5 – x) because we substitute 5 – x in for every x in f(x). Therefore,
f(5 – x) = (5 – x)^2 + 4
= 25 – 10x + x^2 + 4
= x^2 – 10x + 29

To find f(g(-6)), we have to find g(-6) first and substitute that value in for every x in f(x). Therefore,
g(-6) = 5 - (-6) = 11 and f(11) = (11)2 + 4 = 125.
To find f(g(10)), we have to find g(10) first and substitute that value in for every x in f(x). Therefore,
g(10) = 5 – 10 = -5 and f(-5) = (-5)2 + 4 = 29

Example: A meteorologist predicts a low pressure area to move across the region over the next 36 hours. The current temperature of 60 degrees Fahrenheit is forecast to drop 1 degree every 3 hours. What is the composition function that expresses the Celsius temperature as a function of the number of hours from now? Note that C = (5/9)(F – 32).

To solve this we need to know the current temperature and the rate of change of the temperature. We know the current temperature is 60 and there is a 1/3 degree drop expected every hour. We will represent time in hours since the temperature is 60 degrees at t. The temperature in Fahrenheit at time t will be expressed by the function F(t). The temperature at time t expressed in Celsius will be given by the composite function C(F(t)).

F(t) = 60 – (1/3)t since the initial temperature of 60 drops 1/3 degree per hour, represented by (1/3)t.
C(F) = (5/9)(F – 32).
C(F(t)) = (5/9)(60 – (1/3)t – 32)
= (5/9)(28 – (1/3)t) = (140/9) – (5/27)t

What will the temperature be in Fahrenheit and Celsius after the low pressure area moves across the region 36 hours later?

F(36) = 60 – (1/3)(36) = 60 – 12 = 48
C(F(36)) = (140/9) – (5/27)(36) = (140/9) – (180/27) = (420/27) – (180/27) = 240/27 = 8 8/9