Function
Pronunciation: /ˈfʌŋk.ʃən/ Explain
 Figure 1: Function 

A function can be considered like a box with an
input
and an
output.^{[1]}
For each input that goes into the box, a single output comes out of the box. See figure 1.
For any particular input that goes into the box, the same output always
comes out of the box.

 Figure 2: Function f(x)=x^{2} 

In figure 2, the function is labeled y = x^{2}. 'y = x^{2}'
is the rule for transforming
the input into the output. For any number input into the function, square the number
to get the output. Note that every time the number 5 is input into the
function, the number 25 is the output. The input of 5 will generate
only the output of 25.

 Figure 3: Function g(x)={(a,b), (b,c), (c,d), (d,e), (e,a)} 

Functions are defined mathematically as a
set
of
ordered pairs.
The first value in the ordered pair is the input, and the second value is the output. The
ordered pair for the function in figure 3 would be (5,25). 5 is the
input and 25 is the output. For
a function, the first value can not be repeated. Notice this allows
exactly
one output for each input.
The definition of a function as a set of ordered pairs allows us to define functions that
do not use numbers. Take the set {a, b, c, d, e}. If the function
g(x) is defined as g(x) = {(a,b), (b,c), (c,d), (d,e), (e,a)},
one can tell what comes out of the function given
what goes in. If a is the input, b is the output.
For many functions, using an enumerated set of ordered pairs to define the
function is not practical, since there are
infinitely
many inputs and outputs. A rule is typically used in these cases. A rule can
define infinitely many ordered pairs without having to list each ordered pair.

 Figure 4: Function 
 Figure 5: Function 

In the algebra of
real numbers,
functions are defined using
variables.
The input is called the independent variable. The output is
called the dependent variable.
A particular value of the
independent variable,
or input, generates exactly one value of the
dependent variable.
One example of an algebraic function is y = x^{2}. x is
independent variable. x can be any number which can be squared. So x
is independent of anything else. y is the dependent variable. The value of
y depends on the value of x. For example: If x = 5,
y = 25. When x is 5, y can not be anything
but 25. This is because the function states y = x^{2}, so
y must always be the square of x.

Function Notation
 Figure 6: Function notation 

Function notation is commonly used way to write
down functions.
In the example in figure 6, x is the
independent variable, f
is the function name, f(x)
is the dependent variable, and
3x+2 is the rule for
transforming x to f(x). See
also Function Notation.

Domain and Range of Functions
Each function has a
domain
and a
range.
The domain is all the values that the independent variable
can take. See Domain for more information. The
range is all the values that the dependent variable can take.
See Range for more information.
Representations of Functions
Functions can be represented in many ways. Table 1 shows the most common ways.
Representation  Description 
{(1,3), (2,7), (4,5)}  A set of ordered pairs where the first value is the input and the second value is the output. 
f(n) where n is a positive integer and f(n) is the n^{th} Fibonacci number.  A written description. 
y = 3x + 2  An equation with an independent variable and a dependent variable 
f(x) = x^{2}  Function notation 
 A graph where the horizontal axis represents the independent variable and the vertical axis represents the dependent variable. 
 A table where one column or row represents the independent variable. Another column or row represents the dependent variable. 
Table 1: Representations of functions 
Operations on Functions
Functions that are defined over
real numbers
can be added, subtracted, multiplied and divided just like numbers. Also,
functions can be composed.
Operation  Definition  Associativity  Commutativity 
Addition  (f+g)(x) = f(x) + g(x)  (f+(g+h))(x) = ((f+g)+h)(x)  (f+g)(x) = (g+f)(x) 
Subtraction  (fg)(x) = f(x)  g(x)  (f(gh))(x) = ((fg)h)(x)  (fg)(x) ≠ (gf)(x) 
Multiplication  (f·g)(x) = f(x) · g(x)  (f·(g·h))(x) = ((f·g)·h)(x)  (f·g)(x) = (g·f)(x) 
Division  (f÷g)(x) = f(x) ÷ g(x)  (f÷(g÷h))(x) = ((f÷g)÷h)(x)  (f÷g)(x) ≠ (g÷f)(x) 
Composition  (f°g)(x) = f(g(x))  (f°(g°h))(x) = ((f°g)°h)(x)  (f°g)(x) ≠ (g°f)(x) 
Table 2: Summary of operations on functions. 
Addition and Subtraction of Functions
Addition of functions is defined as (f + g)(x) = f(x) + g(x).
To add two functions, add all like terms from each function. Addition of
functions is associative: (f+(g+h))(x) = ((f+g)+h)(x). Addition of
functions is commutative: (f+g)(x) = (g+f)(x).
Step  Equations  Description 
1   These are the functions to add. 
2   Use the definition of function addition to create the new function. 
3   Use the commutative property of addition to place the terms of the new function in order of degree. 
4   Use the distributive property of multiplication over addition and subtraction to start to combine like terms. 
5   Simplify the addition. 
Table 3: Addition of functions 
To subtract two functions, subtract all like terms from each function. The definition
of function subtraction is (fg)(x) = f(x)  g(x). Subtraction of functions
is associative: (f(gh))(x) = ((fg)h)(x). Subtraction of functions
is not commutative: (fg)(x) ≠ (gf)(x).
Step  Equations  Description 
1   These are the functions to subtract. 
2   Use the definition of function subtraction to form the new function. 
3   Distribute negation over the second function. 
4   Use the commutative property of addition to place the terms of the new function in order of degree. 
5   Use the distributive property of multiplication over addition and subtraction to start to combine like terms. 
6   Simplify the addition. 
Table 4: Subtraction of functions 
Multiplication of Functions
To multiply two functions, multiply each term of the first function by each term of the
second function. Function multiplication is defined as
(f·g)(x) = f(x) · g(x). Multiplication of functions is
associative: f(x)·(g(x)·h(x)) = (f(x)·g(x))·h(x).
Multiplication of functions is commutative: f(x)·g(x) = g(x)·f(x).
Step  Equations  Description 
1   These are the functions to multiply. 
2   Use the definition of function multiplication to form the new function. 
3   Multiply each term from the first function times each term from the second function. Since both of these functions are binomials, use the FOIL method. 
4   Simplify the multiplication. 
5   Simplify the parentheses. 
Table 5: Multiplication of functions 
Division of Functions
To divide functions, create a fraction with the
dividend in the numerator and the divisor in the denominator. Simplify the
fraction, if possible. The definition of function division is:
.
Division of functions is associative:
f(x)/(g(x)/h(x)) = (f(x)/g(x))/h(x).
Division of functions is
not commutative:
f(x)/g(x) ≠ g(x)/f(x).
Step  Equations  Description 
1   These are the functions to divide. 
2   Use the definition of function division to form the new function. 
3   Find common factors for the numerator and denominator. 
4   Eliminate common factors. 
5   Rewrite the function using polynomials in standard form. 
Table 6: Division of functions 
Composition of Functions
Composition of functions is taking the function of a function. For example, if
f(x) = x^{2} + 1, and g(x) = x  2, then the composition
of f(x) and g(x), written f°g(x) is
f(g(x)) = (x2)^{2} + 1 = (x^{2}  4x + 2) + 1 = x^{2}4x + 3.
Composition of functions is associative: f°(g°h)(x) = (f°g)°h(x). Composition of
functions is not commutative: f°g(x) ≠ g°f(x).
Step  Equations  Description 
1   These are the functions to compose. 
2   Use the definition of composition to compose the two functions. 
3   Simplify the expression. 
2   Now find g°f. Use the definition of composition to compose the two functions. 
3   Simplify using the distributive property of multiplication over addition and subtraction. 
Table 7: Composition of functions 
References
 McAdams, David E.. All Math Words Dictionary, function. 2nd Classroom edition 201501084799968. pg 83. Life is a Story Problem LLC. January 8, 2015. Buy the book
 Fine, Henry B., Ph. D.. NumberSystem of Algebra Treated Theoretically and Historically. 2nd edition. pg 44. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 7/11/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920pdf#page/n53/mode/1up/search/function. Buy the book
 Goldrei, D.C.. Classic Set Theory: For Guided Independent Study. pp 45. Chapman & Hall Mathematics. July 1, 1996. Last Accessed 7/11/2018. Buy the book
 Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 1225. Thomson, Brooks/Cole. 2005. Last Accessed 7/11/2018. Buy the book
More Information
 McAdams, David E.. Domain. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/d/domain.html.
Cite this article as:
McAdams, David E. Function. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/function.html.
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Revision History
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
7/9/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
2/5/2010: Added "References". (
McAdams, David E.)
3/10/2009: Clarified definition of function notation. (
McAdams, David E.)
2/11/2009: Added section on operations on functions. (
McAdams, David E.)
8/2/2008: Rewrote entire article. (
McAdams, David E.)
8/20/2007: Add this revision history. (
McAdams, David E.)
7/12/2007: Initial version. (
McAdams, David E.)