Graphs of Functions Help

Introduction to Graphs of Functions

It is useful to be able to draw pictures which represent functions. These pictures, or graphs , are a device for helping us to think about functions. In this book we will only graph functions whose domains and ranges are subsets of the real numbers.

We graph functions in the x-y plane. The elements of the domain of a function are thought of as points of the x -axis. The values of a function are measured on the y -axis. The graph of f associates to x the unique y value that the function f assigns to x . In other words, a point ( x , y ) lies on the graph of f if and only if y = f ( x ).

Example 1

Math Note: Notice that for each x in the domain of the function there is one and only one point on the graph—namely the unique point with y value equal to f ( x ). If x is not in the domain of f , then there is no point on the graph that corresponds to x.

Example 2

You Try It: Graph the function y = x + ∣ x ∣.

Example 3

Math Note: A nice, geometrical way to think about the condition that each x in the domain has corresponding to it precisely one y value is this:

If every vertical line drawn through a curve intersects that curve just once, then the curve is the graph of a function.

You Try It: Use the vertical line test to determine whether the locus x ² + y ² = 1 is the graph of a function.

Find practice problems and solutions for these concepts at: Calculus Basics Practice Test.