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We can also give an algebraic expression as the input to a function. For example f ( a + b ) 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.

Function notation

The notation y = f ( x ) defines a function named f . This is read as y is a function of x . The letter x represents the input value, or independent variable. The letter y ,  or f ( x ) , represents the output value, or dependent variable.

Using function notation for days in a month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.

The number of days in a month is a function of the name of the month, so if we name the function f , we write days = f ( month ) or d = f ( m ) . The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

The function 31 = f(January) where 31 is the output, f is the rule, and January is the input.

For example, f ( March ) = 31 , because March has 31 days. The notation d = f ( m ) reminds us that the number of days, d (the output), is dependent on the name of the month, m (the input).

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Interpreting function notation

A function N = f ( y ) gives the number of police officers, N , in a town in year y . What does f ( 2005 ) = 300 represent?

When we read f ( 2005 ) = 300 , we see that the input year is 2005. The value for the output, the number of police officers ( N ) , is 300. Remember, N = f ( y ) . The statement f ( 2005 ) = 300 tells us that in the year 2005 there were 300 police officers in the town.

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Use function notation to express the weight of a pig in pounds as a function of its age in days d .

w = f ( d )

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Instead of a notation such as y = f ( x ), could we use the same symbol for the output as for the function, such as y = y ( x ), 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 f , which is a rule or procedure, and the output y we get by applying f to a particular input x . This is why we usually use notation such as y = f ( x ) , P = W ( d ) , and so on.

Representing functions using tables

A 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.

[link] 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 where D = f ( m ) identifies months by an integer rather than by name.

Month number, m (input) 1 2 3 4 5 6 7 8 9 10 11 12
Days in month, D (output) 31 28 31 30 31 30 31 31 30 31 30 31

[link] defines a function Q = g ( n ) . Remember, this notation tells us that g is the name of the function that takes the input n and gives the output Q  .

Questions & Answers

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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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