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 ) C ( T ) gives the cost C C of heating a house for a given average daily temperature in T T degrees Celsius. The function T ( d ) T ( d ) gives the average daily temperature on day d d of the year. For any given day, Cost = C ( T ( d ) ) 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 ) . T ( d ) . For example, we could evaluate T ( 5 ) 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 ) ) . C ( T ( 5 ) ) .
By combining these two relationships into one function, we have performed function composition, which is the focus of this section.
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 w ( y ) w ( y ) is the wife’s income and h ( y ) h ( y ) is the husband’s income in year y , y , and we want T T to represent the total income, then we can define a new function.
T ( y ) = h ( y ) + w ( y ) T ( y ) = h ( y ) + w ( y )If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
T = h + w T = h + wJust 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 f ( x ) f ( x ) and g ( x ) g ( x ) with real number outputs, we define new functions f + g , f − g , f g , f + g , f − g , f g , and f g f g by the relations
( f + g ) ( x ) = f ( x ) + g ( x ) ( f − g ) ( x ) = f ( x ) − g ( x ) ( f g ) ( x ) = f ( x ) g ( x ) ( f g ) ( x ) = f ( x ) g ( x ) where g ( x ) ≠ 0 ( f + g ) ( x ) = f ( x ) + g ( x ) ( f − g ) ( x ) = f ( x ) − g ( x ) ( f g ) ( x ) = f ( x ) g ( x ) ( f g ) ( x ) = f ( x ) g ( x ) where g ( x ) ≠ 0
Find and simplify the functions ( g − f ) ( x ) ( g − f ) ( x ) and ( g f ) ( x ) , ( g f ) ( x ) , given f ( x ) = x − 1 f ( x ) = x − 1 and g ( x ) = x 2 − 1. g ( x ) = x 2 − 1. Are they the same function?
Begin by writing the general form, and then substitute the given functions.
( g − f ) ( x ) = g ( x ) − f ( x ) ( g − f ) ( x ) = x 2 − 1 − ( x − 1 ) ( g − f ) ( x ) = x 2 − x ( g − f ) ( x ) = x ( x − 1 ) ( g f ) ( x ) = g ( x ) f ( x ) ( g f ) ( x ) = x 2 − 1 x − 1 ( g f ) ( x ) = ( x + 1 ) ( x − 1 ) x − 1 where x ≠ 1 ( g f ) ( x ) = x + 1 ( g − f ) ( x ) = g ( x ) − f ( x ) ( g − f ) ( x ) = x 2 − 1 − ( x − 1 ) ( g − f ) ( x ) = x 2 − x ( g − f ) ( x ) = x ( x − 1 ) ( g f ) ( x ) = g ( x ) f ( x ) ( g f ) ( x ) = x 2 − 1 x − 1 ( g f ) ( x ) = ( x + 1 ) ( x − 1 ) x − 1 where x ≠ 1 ( g f ) ( x ) = x + 1
No, the functions are not the same.
Note: For ( g f ) ( x ) , ( g f ) ( x ) , the condition x ≠ 1 x ≠ 1 is necessary because when x = 1 , x = 1 , the denominator is equal to 0, which makes the function undefined.
Find and simplify the functions ( f g ) ( x ) ( f g ) ( x ) and ( f − g ) ( x ) . ( f − g ) ( x ) .
f ( x ) = x − 1 and g ( x ) = x 2 − 1 f ( x ) = x − 1 and g ( x ) = x 2 − 1Are they the same function?
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:
( f ∘ g ) ( x ) = f ( g ( x ) ) ( f ∘ g ) ( x ) = f ( g ( x ) )We read the left-hand side as “ f “ f composed with g g at x ,” x ,” and the right-hand side as “ f “ f of g g of x . ” x . ” 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 f ( g ( x ) ) ≠ f ( x ) g ( x ) . f ( g ( x ) ) ≠ f ( x ) g ( x ) .
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 g g takes the input x x first and yields an output g ( x ) . g ( x ) . Then the function f f takes g ( x ) g ( x ) as an input and yields an output f ( g ( x ) ) . f ( g ( x ) ) .
In general, f ∘ g f ∘ g and g ∘ f g ∘ f are different functions. In other words, in many cases f ( g ( x ) ) ≠ g ( f ( x ) ) f ( g ( x ) ) ≠ g ( f ( x ) ) for all x . x . We will also see that sometimes two functions can be composed only in one specific order.
For example, if f ( x ) = x 2 f ( x ) = x 2 and g ( x ) = x + 2 , g ( x ) = x + 2 , then
f ( g ( x ) ) = f ( x + 2 ) = ( x + 2 ) 2 = x 2 + 4 x + 4 f ( g ( x ) ) = f ( x + 2 ) = ( x + 2 ) 2 = x 2 + 4 x + 4
g ( f ( x ) ) = g ( x 2 ) = x 2 + 2 g ( f ( x ) ) = g ( x 2 ) = x 2 + 2These expressions are not equal for all values of x , x , so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value x = − 1 2 . x = − 1 2 .
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.
When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input x x and functions f f and g , g , this action defines a composite function , which we write as f ∘ g f ∘ g such that
( f ∘ g ) ( x ) = f ( g ( x ) ) ( f ∘ g ) ( x ) = f ( g ( x ) )The domain of the composite function f ∘ g f ∘ g is all x x such that x x is in the domain of g g and g ( x ) g ( x ) is in the domain of f . f .
It is important to realize that the product of functions f g f g is not the same as the function composition f ( g ( x ) ) , f ( g ( x ) ) , because, in general, f ( x ) g ( x ) ≠ f ( g ( x ) ) . f ( x ) g ( x ) ≠ f ( g ( x ) ) .
Using the functions provided, find f ( g ( x ) ) f ( g ( x ) ) and g ( f ( x ) ) . g ( f ( x ) ) . Determine whether the composition of the functions is commutative .
f ( x ) = 2 x + 1 g ( x ) = 3 − x f ( x ) = 2 x + 1 g ( x ) = 3 − xLet’s begin by substituting g ( x ) g ( x ) into f ( x ) . f ( x ) .
f ( g ( x ) ) = 2 ( 3 − x ) + 1 = 6 − 2 x + 1 = 7 − 2 x f ( g ( x ) ) = 2 ( 3 − x ) + 1 = 6 − 2 x + 1 = 7 − 2 x
Now we can substitute f ( x ) f ( x ) into g ( x ) . g ( x ) .
g ( f ( x ) ) = 3 − ( 2 x + 1 ) = 3 − 2 x − 1 = −2 x + 2 g ( f ( x ) ) = 3 − ( 2 x + 1 ) = 3 − 2 x − 1 = −2 x + 2
We find that g ( f ( x ) ) ≠ f ( g ( x ) ) , g ( f ( x ) ) ≠ f ( g ( x ) ) , so the operation of function composition is not commutative.
The function c ( s ) c ( s ) gives the number of calories burned completing s s sit-ups, and s ( t ) s ( t ) gives the number of sit-ups a person can complete in t t minutes. Interpret c ( s ( 3 ) ) . c ( s ( 3 ) ) .
The inside expression in the composition is s ( 3 ) . s ( 3 ) . Because the input to the s-function is time, t = 3 t = 3 represents 3 minutes, and s ( 3 ) s ( 3 ) is the number of sit-ups completed in 3 minutes.
Using s ( 3 ) s ( 3 ) as the input to the function c ( s ) c ( s ) gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).
Suppose f ( x ) f ( x ) gives miles that can be driven in x x hours and g ( y ) g ( y ) gives the gallons of gas used in driving y y miles. Which of these expressions is meaningful: f ( g ( y ) ) f ( g ( y ) ) or g ( f ( x ) ) ? g ( f ( x ) ) ?
The function y = f ( x ) y = f ( x ) is a function whose output is the number of miles driven corresponding to the number of hours driven.
number of miles = f ( number of hours ) number of miles = f ( number of hours )The function g ( y ) g ( y ) is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:
number of gallons = g ( number of miles ) number of gallons = g ( number of miles )The expression g ( y ) g ( y ) takes miles as the input and a number of gallons as the output. The function f ( x ) f ( x ) requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression f ( g ( y ) ) f ( g ( y ) ) is meaningless.
The expression f ( x ) f ( x ) takes hours as input and a number of miles driven as the output. The function g ( y ) g ( y ) requires a number of miles as the input. Using f ( x ) f ( x ) (miles driven) as an input value for g ( y ) , g ( y ) , where gallons of gas depends on miles driven, does make sense. The expression g ( f ( x ) ) g ( f ( x ) ) makes sense, and will yield the number of gallons of gas used, g , g , driving a certain number of miles, f ( x ) , f ( x ) , in x x hours.
Are there any situations where f ( g ( y ) ) f ( g ( y ) ) and g ( f ( x ) ) g ( f ( x ) ) would both be meaningful or useful expressions?
Yes. For many pure mathematical functions, both compositions make sense, even though they usually produce different new functions. In real-world problems, functions whose inputs and outputs have the same units also may give compositions that are meaningful in either order.
The gravitational force on a planet a distance r from the sun is given by the function G ( r ) . G ( r ) . The acceleration of a planet subjected to any force F F is given by the function a ( F ) . a ( F ) . Form a meaningful composition of these two functions, and explain what it means.
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.
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.
Using Table 1, evaluate f ( g ( 3 ) ) f ( g ( 3 ) ) and g ( f ( 3 ) ) . g ( f ( 3 ) ) .