When one quantity depends on another, we have a function. Lets look at some examples of functions:

1. The human population, P depends on the time t . If we were to gather the population for each year for the past ten years, then P(t) or P(2000) should give us the population of the world for that year. Thus, we say that P is a functon of time.
2. The cost to ship a package is a function of w, the weight. Each company has their own formula for calculating the cost when weight is given.

A Function f is a rule that assigns exactly one element, called f(x) to each element x
The set of x, lets call it D, is called the domain of the function. The functon f(x) means value of f at x. It is read as "f of x". The range of the function is the set of all possible values for f(x) as x varies throughout the domain.

Lets try solving a function.
Example 1. if [math]f(x) = 2x^2 - 5x +1 [/math] and h [math]\not= 0[/math], evaluate [math] \frac{f(a+h)-f(a)} {h}[/math]
Solution:  To solve, we began with evaluating f(a+h) by replacing x by a+h in f(x):
[math]f(a+h)[/math]=[math]2(a+h)^2 - 5(a+h) + 1[/math]
         =[math]2a^2 +4ah +2h^2 -5a -5h +1[/math]
Then we subsitute into the given expression and simplify:
[math]\frac {f(a+h)-f(a)} {h} = \frac {(2(a+h)^2 - 5(a+h) + 1) - (2a^2 - 5a + 1)} {h}
           =\frac{4ah + 2h^2 - 5h}{h} = 4a+2h-5 [/math]
 

Now lets do another example involving range and domain of a function
 Example 2. Find the domain of each function:
[math]f(x) = \sqrt{x+2}[/math]
and
[math]g(x) = \frac {1}{x^2 - x} [/math]