6.5 Function Composition
Suppose we wanted to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The temperature depends on the day, and the cost depends on the temperature. Using descriptive variables, we can notate these two functions.
The first function, C(T), gives the cost C of heating a house when the average daily temperature is T degrees Celsius, and the second, T(d), gives the average daily temperature on day d of the year in some city. If we wanted to determine the cost of heating the house on the 5th day of the year, we could do this by linking our two functions together, an idea called composition of functions. Using the function T(d), we could evaluate T(5) to determine the average daily temperature on the 5th day of the year. We could then use that temperature as the input to the C(T) function to find the cost to heat the house on the 5th day of the year: C(T(5)).
Composition of Functions
When the output of one function is used as the input of another, we call the entire operation a composition of functions. We write , and read this as “f of g of x” or “f composed with g at x”.
An alternate notation for composition uses the composition operator:
is read “f of g of x” or “f composed with g at x”, just like
Be careful! . It is NOT a multiplication! and it is NOT multiplied by .
Example of a Function Composition
Suppose c(s) gives the number of calories burned doing s sit-ups, and s(t) gives the number of sit-ups a person can do in t minutes. Interpret c(s(3)).
When we are asked to interpret, we are being asked to explain the meaning of the expression in words. The inside expression in the composition is s(3). Since the input to the s function is time, the 3 is representing 3 minutes, and s(3) is the number of sit-ups that can be done in 3 minutes. Taking this output and using it as the input to the c(s) function will gives us the calories that can be burned by the number of sit-ups that can be done in 3 minutes.
Note that it is not important that the same variable be used for the output of the inside function and the input to the outside function. However, it is essential that the units on the output of the inside function match the units on the input to the outside function, if the units are specified.
Another Example of Function Composition
Suppose gives miles that can be driven in hours, and gives the gallons of gas used in driving y miles. Which of these expressions is meaningful: or ?
The expression takes miles as the input and outputs a number of gallons. The function is expecting a number of hours as the input; trying to give it a number of gallons as input does not make sense. Remember the units must match, and number of gallons does not match number of hours, so the expression is meaningless.
The expressi