Functions Part II

In the previous section, we learned that functions passed the vertical line test. There is another way to classify functions which pass the horizontal line test. A function passes the horizontal line test if we draw a horizontal line anywhere in the domain of , and that line crosses in at most one or less points. If a function passes the horizontal line test then we call it one-to-one, or 1-1.

In addition to being classified as one-to-one, we can call a polynomial if can be written in the form

.

Each a, except the last one, in the above notation is a coefficient to x. Each a can be any real number. The n's are the exponent to x. They determine the degree of the polynomial. If n = 0, then we would have something like f(x) = 5. The five would be a constant which means there is no variable associated with it. If n = 1 then we would have something like f(x) = 5x + 1. In this case is linear. If n = 2 we would have something like . In that case is quadratic. In this example, there is no constant or linear term, but when classifying polynomials, the term with the greatest degree determines the classification.

There is one last classification of functions we are going to talk about. A function is an even function if f(x)=f(-x). This might be impossible you say, but take the function we just worked with,
is an even function. x squared is equal -x squared. So f(x) is an even function. A function is odd if f(-x)=-f(x). Take for example the function f(x) = x.

.

f(x) is now odd.