Functions

       There are times when various quantities can be related to one another. If this relationship follows a regular pattern then we can assign variables to these quantities and write a mathematical equation to express this relationship. This relationship is called a function. If this relationship does not follow a regular pattern, we can still assign variables to these quantities, but we cannot write an equation to express this relationship.

       Suppose we talk about the squaring relationship, i.e., 1 squared is 1, 2 squared is 4, 3 squared is 9, etc. We can show this relationship using set notation like this: { . . ., (1,1), (2,4), (3,9), . . .}. 1 is related to 1, 2 is related to 4, 3 is related to 9, etc. The ellipsis on either side indicates the pattern repeats indefinitely to the right and to the left. This is a regular pattern and we can therefore write an equation like this: let 'x' represent the first number and 'y' represent the second number then we have y = x². If we're not too concerned naming the second number, y, we may employ functional notation. We only need a name for the function, so let's call the function 'f'. The equation becomes f(x) = x². And we read this equation as “f of x equals x squared.”

       Function names are not limited to single letters, but single letters are used for brevity. We could write square(x) = x², and this would be quite alright, and we'd be talking about the square function.

For an explanation of independent and dependent variables and domains and ranges and maps and graphs

see : http://www.k12math.com/math-concepts/graphs.htm

       Substituting into a function refers to taking the value for the independent variable and replacing all terms named by the variable with that value then simplifying the result. This result will be the value of the dependent variable.

Example: f(x) = 2·x – x² + 13

1. for x = 1 f(1) = 2·1 – 1² + 13 = 14

2. for x = 0 f(0) = 2·0 – 0² + 13 = 15

3. for x = -2 f(-2) = 2(-2) – (-2)² + 13 = -4 – (4) + 13 = 5

     We can also ask: if f(x) = -2, what are the values of x?

     So, we have -2 = 2x – x² + 13

     0 = 2x – x² + 15

     0 = (x + 3) ( -x + 5 )

     The solution is x = -3 or x = 5 x ε { -3, 5 }

(Remember, quadratic equations have 2 solutions.)

       Functions can be described by the way they map the independent variables to the dependent variables. In the previous example we have two values of x that produce the same value of y, namely -3 and 5 map to -2. In this case we say that this function, f, is a many-to-one mapping. Suppose we have the function g(x) = x^½. For x = 4 the square root of 4 is +2 and -2. In this case one value for x gives two values for y. This function, g, is a one-to-many mapping. Now suppose we have h(x) = 2x. In this case for every value of x is is a unique value for y. This function, h, is a one-to-one mapping. It is possible to have functions that are many-to-many mappings, but many of these functions cannot be described by an equation, but are usually described with conditional set notation relating the dependent variable to the independent variable(s).

     Functions can be combined arithmetically (as long as domain constraints are named as necessary.) For example:

suppose f(x) = 3x + 2 and g(x) = x – 10

where the Domain and Range = Real Numbers

1. f(x) + g(x)  = (3x + 2) + (x – 10) = 4x – 8

2. f(x) – g(x)  = (3x + 2) – (x – 10) = 2x + 12

3. f(x) · g(x)   = (3x + 2) · (x – 10) = 3x² – 28x – 20

4. f(x) / g(x)   = (3x + 2)/(x – 10)

In #4 above, notice that if x = 10 we'd be dividing by 0, so our domain constraint would be x ≠ 10.

      One last operation we'll perform with functions is called composition of functions. This operation is akin to substituting values for the independent variable into a function, instead we substitute another function as an independent variable value into that first variable's function. An example will help.

Using our previous functions f and g, let's substitute g into f. We get:

f( g(x) ) = 3( g(x) ) + 2

f( g(x) ) = 3( x – 10 ) + 2

f( g(x) ) = 3x – 28

f º g = 3x – 28

note the last equation, “f of g of x” is notated as f º g

let's do another:

g º f = ( f(x) ) – 10

= (3x + 2 ) – 10

= 3x – 8