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3.4+Composition of Functions 239 47. The graph in Figure 19 illustrates the decay of a radioactive substance over t days. a 5 10 15 20 Time (days) Figure 19 Use the graph to estimate the average decay rate fromt=Stot=15. 3.4 Composition of Functions Learning Objectives In this section, you will: > Combine functions using algebraic operations. > Create a new function by composition of functions. > Evaluate composite functions. > Find the domain of a composite function. > Decompose a composite function into its component functions. Suppose we want 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 in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day. Using descriptive variables, we can notate these two functions. The function C (T) gives the cost C of heating a house for a given average daily temperature in T degrees Celsius. The function T (d) gives the average daily temperature on day d of the year. For any given day, Cost = C (T (d)) means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature T (d) . For example, we could evaluate T (5) to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would write C (T (5)). Cost for the temperature SI c(T(5)) Temperature on day 5 By combining these two relationships into one function, we have performed function composition, which is the focus of this section. Combining Functions Using Algebraic Operations Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function. Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that 
