# 1.4 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

#### Questions & Answers

what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply

### Read also:

#### Get the best Precalculus course in your pocket!

Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

 By By By By