Recall that a function is like a machine: given an input [math]x[/math] the function returns a value [br][math]f\left(x\right)[/math]. A function of two variables operates similarly: given an input [math]\left(x,y\right)[/math] the function returns a value [math]f\left(x,y\right)[/math].

When working with a function of two variables, it may be written as [math]z=f\left(x,y\right)[/math]. We refer to [math]x[/math] and [math]y[/math] as the [b]independent variables [/b]and [math]z[/math] as the [b]dependent variable[/b]. The set of ordered pairs [math]\left(x,y\right)[/math] that produce an output is the [b]domain[/b] of the function and the outputs [math]f\left(x,y\right)[/math] that correspond to them make up the [b]range[/b].

Given the function [math]z=f\left(x,y\right)[/math], its [b]level curves[/b] are the two-dimensional curves we get by setting [math]z=k[/math] (where [math]k[/math] is a constant). The equations of the level curves are [math]f\left(x,y\right)=k[/math]. [br]

In the applet below, the left shows the level curves on the [math]xy-[/math]plane while the right shows the surface [math]z=k[/math].[br][br]To explore the level curves for different functions, choose an example and change the value of [math]k[/math], using the slider, to see different level curves.