Definition of a Function

A function is a relation is which for each element in the domain thee corresponds exactly one element in the range.

All functions are relations, but not all relations are functions. A function is a special relation in which no two ordered pairs have the same first coordinate.

For example, given the relation { (3,5), (4,2), (3,6), (5,7)} , there are two ordered pairs, (3,5) and (3,6), with the same first coordinate. So, this relation is not a function.

Example. Determine whether or not the following relation is a function:

{ (x,y) | y = x + 5}

Solution. The ordered pairs of this relation are found by choosing values for x, and substituting them into the equation, y = x + 5, to obtain the corresponding y. For each x value, there is only one possible "answer" for y. So there could be no two ordered pairs with the same x-coordinate. Thus, the given relation is a function.

The graph of a relation is the set of points in the plane that correspond to the ordered pairs of the relation. You can determine whether a given graph is the graph of a function by using the Vertical Line Test.

Vertical Line Test

If each vertical line intersects a graph at no more than one point, the graph is the graph of a function.

In other words, if a vertical line intersects a graph at more than one point, the graph is not the graph of a function.

Function Notation

Consider the function defined by the equation y = x + 5. We use function notation to name the function in the following way: f(x) = x + 5.

The notation f(x) is read "f of x". f is the name of the function, and x is an element of the domain of f. The value of the function corresponding to x is f(x).

Also, f(x) is just another notation for y. You can always replace f(x) with y or y with f(x).

Most of the time, the letters f, g, h, and k are used to name functions. Examples of functions are:

k(x) = , g(t) = t- 4t + 7, and h(a) =

Note: The parentheses in the notation f(x) do not indicate multiplication. f(x) is "f of x", not "f times x".

To evaluate a function, we determine the value of f(x) for a specific value of x. f(a) is the value of f(x) when a is substituted for x in f(x).

Example. a) Given f(x) = , find f(-4).

b) Given h(x) = | 3 – 2x| , find h(5).

Solution. a) f(x) =

f(-4) =

=

= -18

b) h(x) = | 3 – 2x|

h(5) = | 3 – 2(5)|

= | 3 – 10|

= | –7 |

= 7