A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first. Show
{(1,2),(2,4),(3,6) ,(4,8),(5,10)}\left\{\left(1,2\right),\left(2,4\right),\left(3,6\right),\left(4,8\right),\left(5,10\right)\right\} The domain is {1,2,3,4,5}\left\{1,2,3,4,5\right\} . The range is {2,4,6,8,10}\left\{2,4,6,8,10\right\} . Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter xx . Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter yy . A function ff is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, {1,2,3,4 ,5}\left\{1,2,3,4,5\right\} , is paired with exactly one element in the range, {2,4,6,8,10 }\left\{2,4,6,8,10\right\} . Now let’s consider the set of ordered pairs that relates the terms "even" and "odd" to the first five natural numbers. It would appear as {(odd,1),(even,2),(odd,3),(even,4 ),(odd,5)}\left\{\left(\text{odd},1\right),\left(\text{even},2\right),\left(\text{odd},3\right),\left(\text{even},4\right),\left(\text{odd},5\right)\right\} Notice that each element in the domain, {even,odd}\left\{\text{even,}\text{odd}\right\} is not paired with exactly one element in the range, { 1,2,3,4,5}\left\{1,2,3,4,5\right\} . For example, the term "odd" corresponds to three values from the domain, {1 ,3,5}\left\{1,3,5\right\} and the term "even" corresponds to two values from the range, {2,4}\left\{2,4\right\} . This violates the definition of a function, so this relation is not a function. Figure 1 compares relations that are functions and not functions. Figure 1. (a) This relationship is a function because each input is associated with a single output. Note that inputq q and rr both give output nn . (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input is associated with two different outputs. A General Note: FunctionA function is a relation in which each possible input value leads to exactly one output value. We say "the output is a function of the input." The input values make up the domain, and the output values make up the range. How To: Given a relationship between two quantities, determine whether the relationship is a function.
Example 1: Determining If Menu Price Lists Are FunctionsThe coffee shop menu, shown in Figure 2 consists of items and their prices.
Solution
Example 2: Determining If Class Grade Rules Are FunctionsIn a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? The table below shows a possible rule for assigning grade points.
SolutionFor any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average. In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average. Try It 1The table below lists the five greatest baseball players of all time in order of rank.
a) Is the rank a function of the player name? b) Is the player name a function of the rank? Solution Using Function NotationOnce we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions. To represent "height is a function of age," we start by identifying the descriptive variables hh for height andaa for age. The lettersf,gf,g , andhh are often used to represent functions just as we usex,yx,y , andzz to represent numbers andA,BA,B , andCC to represent sets.{h is f of aWe name the function f; height is a function of age.h=f(a) We use parentheses to indicate the function input. f (a)We name the function f; the expression is read as "f of a."\begin{cases}h\text{ is }f\text{ of }a\qquad & \qquad & \qquad & \qquad & \text{We name the function }f;\text{ height is a function of age}.\qquad \\ h=f\left(a\right)\qquad & \qquad & \qquad & \qquad & \text{We use parentheses to indicate the function input}\text{. }\qquad \\ f\left(a\right)\qquad & \qquad & \qquad & \qquad & \text{We name the function }f;\text{ the expression is read as "}f\text{ of }a\text{."}\qquad \end{cases} Remember, we can use any letter to name the function; the notation h(a)h\left(a\right) shows us thathh depends onaa . The valueaa must be put into the functionhh to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.We can also give an algebraic expression as the input to a function. For example f(a+b)f\left(a+b\right) means "first add a and b, and the result is the input for the function f." The operations must be performed in this order to obtain the correct result.A General Note: Function NotationThe notation y=f(x)y=f\left(x\right) defines a function named ff . This is read as "y"y is a function of x."x." The letter xx represents the input value, or independent variable. The letter y,y\text{,}{\hspace{0.17em}} or f(x)f\left(x\right) , represents the output value, or dependent variable. Example 3: Using Function Notation for Days in a MonthUse function notation to represent a function whose input is the name of a month and output is the number of days in that month. SolutionThe number of days in a month is a function of the name of the month, so if we name the function ff , we write days=f(month)\text{days}=f\left(\text{month}\right) or d=f(m)d=f\left(m\right) . The name of the month is the input to a "rule" that associates a specific number (the output) with each input. Figure 4For example, f(March)=31f\left(\text{March}\right)=31 , because March has 31 days. The notation d =f(m)d=f\left(m\right) reminds us that the number of days, dd (the output), is dependent on the name of the month, mm (the input). Example 4: Interpreting Function NotationA function N=f(y)N=f\left(y\right) gives the number of police officers, NN , in a town in year yy . What does f(2005)=300f\left(2005\right)=300 represent? SolutionWhen we read f(2005)=300f\left(2005\right)=300 , we see that the input year is 2005. The value for the output, the number of police officers (N)\left(N\right) , is 300. Remember, N=f(y) N=f\left(y\right) . The statement f(2005)=300f\left(2005\right)=300 tells us that in the year 2005 there were 300 police officers in the town. Q & AInstead of a notation such as y=f(x)y=f\left(x\right) , could we use the same symbol for the output as for the function, such asy=y(x)y=y\left(x\right) , meaning "y is a function of x?"Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as ff , which is a rule or procedure, and the outputyy we get by applyingff to a particular inputx x y=f(x),P=W(d)y=f\left(x\right),P=W\left(d\right) , and so on.Representing Functions Using TablesA common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship. The table below lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function f f whereD=f(m)D=f\left(m\right) identifies months by an integer rather than by name.
The table below defines a function Q=g(n)Q=g\left(n\right) . Remember, this notation tells us thatgg is the name of the function that takes the inputnn and gives the outputQ. Q\text{.}{\hspace{0.17em}}
The table below displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.
How To: Given a table of input and output values, determine whether the table represents a function.
Example 5: Identifying Tables that Represent FunctionsWhich table, a), b), or c), represents a function (if any)? a) Table A
b) Table B
c) Table C
Solutiona) and b) define functions. In both, each input value corresponds to exactly one output value. c) does not define a function because the input value of 5 corresponds to two different output values. When a table represents a function, corresponding input and output values can also be specified using function notation. The function represented by a) can be represented by writing f(2)=1,f(5)=3,and f(8)=6f\left(2\right)=1,f\left(5\right)=3,\text{and }f\left(8\right)=6 Similarly, the statements g(−3) =5,g(0)=1,and g(4)=5g\left(-3\right)=5,g\left(0\right)=1,\text{and }g\left(4\right)=5 represent the function in b). c) cannot be expressed in a similar way because it does not represent a function. Licenses and AttributionsCC licensed content, Shared previously
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How do I know if a table is a function or not?How do you figure out if a relation is a function? You could set up the relation as a table of ordered pairs. Then, test to see if each element in the domain is matched with exactly one element in the range. If so, you have a function!
What makes a function on a table?A function table is a visual table with columns and rows that displays the function with regards to the input and output. Younger students will also know function tables as function machines. Every function has a rule that applies and represents the relationships between the input and output.
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