# Functions

## Composition of 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 
 gives the cost 
 of heating a house for a given average daily temperature in 
 degrees Celsius. The function 
 gives the average daily temperature on day 
 of the year. For any given day, 
 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 
 For example, we could evaluate 
 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 


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 year’s incomes and then collecting all the data in a new column. If 
 is the wife’s income and 
 is the husband’s income in year 
 and we want 
 to represent the total income, then we can define a new function.
If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.
For two functions 
 and 
 with real number outputs, we define new functions 
 and 
 by the relations

### Create a Function by Composition of Functions

Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:
We read the left-hand side as 
  composed with 
 at 
  and the right-hand side as 
  of 
 of 
  The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol 
 is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases 
 

It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function 
 takes the input 
 first and yields an output 
 Then the function 
 takes 
 as an input and yields an output 


In general, 
 and 
 are different functions. In other words, in many cases 
 for all 
 We will also see that sometimes two functions can be composed only in one specific order.
For example, if 
 and 
 then
but
These expressions are not equal for all values of 
 so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value 


Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.

### Evaluating Composite Functions

Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.

### Evaluating Composite Functions Using Tables

When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.

### Evaluating Composite Functions Using Graphs

When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the 
 and 
 axes of the graphs.

### Evaluating Composite Functions Using Formulas

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition 
 To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like 
 we substitute the value inside the parentheses into the formula wherever we see the input variable.

### Finding the Domain of a Composite Function

As we discussed previously, the domain of a composite function such as 
 is dependent on the domain of 
 and the domain of 
 It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as 
 Let us assume we know the domains of the functions 
 and 
 separately. If we write the composite function for an input 
 as 
 we can see right away that 
 must be a member of the domain of 
 in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that 
 must be a member of the domain of 
 otherwise the second function evaluation in 
 cannot be completed, and the expression is still undefined. Thus the domain of 
 consists of only those inputs in the domain of 
 that produce outputs from 
 belonging to the domain of 
 Note that the domain of 
 composed with 
 is the set of all 
 such that 
 is in the domain of 
 and 
 is in the domain of 



### Decomposing a Composite Function into its Component Functions

In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.

### Key Equation


### Key Concepts

1. We can perform algebraic operations on functions. See .
2. When functions are composed, the output of the first (inner) function becomes the input of the second (outer) function.
3. The function produced by composing two functions is a composite function. See  and .
4. The order of function composition must be considered when interpreting the meaning of composite functions. See .
5. A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
6. A composite function can be evaluated from a table. See .
7. A composite function can be evaluated from a graph. See .
8. A composite function can be evaluated from a formula. See .
9. The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See  and .
10. Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
11. Functions can often be decomposed in more than one way. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, use each pair of functions to find 
 and 
 Simplify your answers.
For the following exercises, use each set of functions to find 
 Simplify your answers.
For the following exercises, find functions 
 and 
 so the given function can be expressed as 



### Graphical

For the following exercises, use the graphs of 
 shown in , and 
 shown in , to evaluate the expressions.
For the following exercises, use graphs of 
 shown in , 
 shown in , and 
 shown in , to evaluate the expressions.

### Numeric

For the following exercises, use the function values for 
 shown in  to evaluate each expression.
For the following exercises, use the function values for 
 shown in  to evaluate the expressions.
For the following exercises, use each pair of functions to find 
 and 


For the following exercises, use the functions 
 and 

to evaluate or find the composite function as indicated.

### Extensions

For the following exercises, use 
 and 


For the following exercises, let 
 
 and 


For the following exercises, find the composition when 
 for all 
 and 



### Real-World Applications
