# 3.4 Composition of functions

 Page 1 / 9
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 $\text{\hspace{0.17em}}C\left(T\right)\text{\hspace{0.17em}}$ gives the cost $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of heating a house for a given average daily temperature in $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ degrees Celsius. The function $\text{\hspace{0.17em}}T\left(d\right)\text{\hspace{0.17em}}$ gives the average daily temperature on day $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ of the year. For any given day, $\text{\hspace{0.17em}}\text{Cost}=C\left(T\left(d\right)\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}T\left(d\right).\text{\hspace{0.17em}}$ For example, we could evaluate $\text{\hspace{0.17em}}T\left(5\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}C\left(T\left(5\right)\right).$

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 $\text{\hspace{0.17em}}w\left(y\right)\text{\hspace{0.17em}}$ is the wife’s income and $\text{\hspace{0.17em}}h\left(y\right)\text{\hspace{0.17em}}$ is the husband’s income in year $\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}$ and we want $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ to represent the total income, then we can define a new function.

$T\left(y\right)=h\left(y\right)+w\left(y\right)$

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 $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ with real number outputs, we define new functions $f+g,\text{\hspace{0.17em}}f-g,\text{\hspace{0.17em}}fg,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{f}{g}\text{\hspace{0.17em}}$ by the relations

sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
Wrong question
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb