Functions Part I

To illustrate the idea of a function let us define our line with the equation . Graphically, a function is a rule that assigns to each x-coordinate defined exactly one y-coordinate. So a line containing the points (2,3) and (2,-3) would not be the graph of a function. An easy way to find out if the graph of an equation is a function is to use the vertical line test. If a perfectly vertical line crosses the graph of the given equation at no more than one point, then the equation is function.

Let's go back to our example. We can see the graph of in the following graph:

If we draw a vertical line across any x, the line will cross y at exactly one point. So our line is function. Since it is a function, we can write:. One more final note on the definition of a function- the set of x the function takes values on is known as the domain. The set of y values that the function takes on is the range. This is not a rigorous definition of a function, but will suffice for the time being.

We can also plug one function into another in what is known as the composition of functions. Suppose we have two functions, and . The composition the two functions: f circle g would look like this:


What this means in terms of the two functions we have just defined, and , is that we put whatever is on the right side of , into each in . What this would look like with the actual numbers is:


The composition reversed would like: