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Function Notation

Module by: Kenny M. Felder. E-mail the author

Summary: This module describes notation for functions.

Function Notation

Functions are represented in math by parentheses. When you write f(x)f(x) size 12{f \( x \) } {} you indicate that the variable ff size 12{f} {} is a function of—or depends on—the variable xx size 12{x} {}.

For instance, suppose f(x)=x2+3xf(x)=x2+3x size 12{f \( x \) =x rSup { size 8{2} } +3x} {} . This means that f is a function that takes whatever you give it, and squares it, and multiplies it by 3, and adds those two quantities.

Table 1
7 10 x y a dog 7 10 x y a dog x-squared plus 3x Gearbox f ( 7 ) = 7 2 + 3 ( 7 ) = 70 f ( 10 ) = 10 2 + 3 ( 10 ) = 130 f ( x ) = x 2 + 3x f ( y ) = y 2 + 3y f ( dog ) = ( dog ) 2 + 3 ( dog ) ( *not in the domain ) f ( 7 ) = 7 2 + 3 ( 7 ) = 70 f ( 10 ) = 10 2 + 3 ( 10 ) = 130 f ( x ) = x 2 + 3x f ( y ) = y 2 + 3y f ( dog ) = ( dog ) 2 + 3 ( dog ) ( *not in the domain )

The notation f(7)f(7) size 12{f \( 7 \) } {} means “plug the number 7 into the function ff size 12{f} {}.” It does not indicate that you are multiplying ff size 12{f} {} times 7. To evaluate f(7)f(7) size 12{f \( 7 \) } {} you take the function f(x)f(x) size 12{f \( x \) } {} and replace all occurrences of the variable x with the number 7. If this function is given a 7 it will come out with a 70.

If we write f(y)=y2+3yf(y)=y2+3y size 12{f \( y \) =y rSup { size 8{2} } +3y} {} we have not specified a different function. Remember, the function is not the variables or the numbers, it is the process. f(y)=y2+3yf(y)=y2+3y size 12{f \( y \) =y rSup { size 8{2} } +3y} {} also means “whatever number comes in, square it, multiply it by 3, and add those two quantities.” So it is a different way of writing the same function.

Just as many students expect all variables to be named xx size 12{x} {}, many students—and an unfortunate number of parents—expect all functions to be named ff size 12{f} {}. The correct rule is that—whenever possible—functions, like variables, should be named descriptively. For instance, if Alice makes $100/day, we might write:

  • Let m equal the amount of money Alice has made (measured in dollars)
  • Let t equal the amount of time Alice has worked (measured in days)
  • Then, m(t)=100tm(t)=100t size 12{m \( t \) ="100"t} {}

This last equation should be read “ mm size 12{m} {} is a function of tt size 12{t} {} (or mm size 12{m} {} depends on tt size 12{t} {}). Given any value of the variable tt size 12{t} {}, you can multiply it by 100 to find the corresponding value of the variable mm size 12{m} {}.”

Of course, this is a very simple function! While simple examples are helpful to illustrate the concept, it is important to realize that very complicated functions are also used to model real world relationships. For instance, in Einstein’s Special Theory of Relativity, if an object is going very fast, its mass is multiplied by 11v29101611v291016 size 12{ { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {9 cdot "10" rSup { size 8{"16"} } } } } } } } {}. While this can look extremely intimidating, it is just another function. The speed vv size 12{v} {} is the independent variable, and the mass mm size 12{m} {} is dependent. Given any speed vv size 12{v} {} you can determine how much the mass mm size 12{m} {} is multiplied by.

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