In an equation, an independent variable is a variable whose value does not depend on the value of other variables. In algebra, this is often the variable. The independent variable generally appears as part of an expression on one side of the equation, as with .
In an equation, a dependent variable is a variable whose value depends on the the values of one or more other variables. In algebra, this is usually the variable. The dependent variable is often isolated on one side of an equation, as with .
A function relates an independent variable, the input of the function, to a dependent variable, the output of the function so that for each value of the independent variable, there is only one value of the dependent variable. If we denote a function as
There are many types of functions - polynomial functions (which include a wide variety of functions), rational functions, absolute value functions, square root functions, and more.
We can use the vertical line test to determine whether or not a graph represents a function. Since a function can only have one output for each input, or one value of the dependent variable for each value of the independent variable, its graph can only cross each horizontal position on the coordinate plane at most one time. To perform the vertical line test on a graph, take a straight object, like a pencil or a ruler, position it so that it is parallel with the vertical axis of the graph, and then move it horizontally from one side of the graph to the other. If at any position the vertical line intersects the graph more than once, the graph does not represent a function. Please note that use of the vertical line test assumes that the independent variable,, is represented on the horizontal axis and that the dependent variable, , is represented on the vertical axis.