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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 ) ) .

Explanation of C(T(5)), which is the cost for the temperature and T(5) is the 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 year’s incomes and then collecting all the data in a new column. If w ( y ) is the wife’s income and h ( y ) is the husband’s income in year y , and we want T to represent the total income, then we can define a new function.

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

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 f ( x ) and g ( x ) with real number outputs, we define new functions f + g , f g , f g , and 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

Questions & Answers

Solve the problem 3x^2_2x+4=0
Ayoman Reply
can anyone recommend an app or. website to help me refresh my knowledge and abilities? I stopped learning more about calc and trig in the 90's
scott Reply
if 6x=-2 find value of 6/x
issaya Reply
l don't know
Aman
I don't know
Aman
l don't know
Aman
I don't know
Aman
x=.333333333334, so 6/x= 18±
scott
x=.33333333334± 6/x=18±
scott
that is strange, I don't remember inserting the À. the result I got was x= 3333333334± 6/x=18±
scott
I need glasses
scott
X=(-1÷3) X equals minus one third
Melvin
if sin15°=√p, express the following in terms of p
Norman Reply
I am learning and does someone have the time to tell me what or where this particular equation would be used?
scott
prove sin²x+cos²x=3+cos4x
Kiddy Reply
the difference between two signed numbers is -8.if the minued is 5,what is the subtrahend
jeramie Reply
the difference between two signed numbers is -8.if the minuend is 5.what is the subtrahend
jeramie
what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
ask
Ans
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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