Functions, Limits, Continuity


Brief Review of Functions

Summary Brief Review of Functions
Figure %:Graph of the function y = x3

We can see that it is a function because it passes the vertical line test. We can also see that it assigns only one x value to each y value. Thus, it is a one-to-one function. Again from precalculus, we can see graphically whether a function is a one-to-one function by using the horizontal line test:

Figure %:Horizontal line test on the functions y = x3 and y = x2 + 2

Any horizontal line we draw through the graph of the function y = x3 passes through only one point, so it must be assigning only one x value to each y, and can therefore be considered a one-to-one function. Horizontal lines through y = x2 + 2 pass through more than one point, so this function fails the horizontal line test.

In summary, for a rule to be a function, its graph must pass the vertical line test. To be a one-to-one function, it must pass both the vertical line test and the horizontal line test.

Functional Notation

In this guide, we will often give functions names such as f (x), g(x), h(x), etc. For example, when we say "f (x) = x2 + 2", we mean for f (x) to refer to the rule that assigns the number y = x2 + 2 to any real number x.

Two Types of Functions: Rational and Polynomial

As we proceed, two types of functions to be aware of are polynomial functions and rational functions.

Polynomial Functions

A polynomial function is any function of the form