[list][*]Use the input boxes for [math]g(x)[/math] and [math]f(x)[/math] to define the "inside" and "outside" function for the function composition. [/*][*]Use the input box / slider tool for [math]c[/math] to adjust the input for the inside function. Observe that the output of the inside function becomes the input of the outside function. [/*][*]Use the "Trace f(g(x))" button to trace out the graph of the composite function [math]y=f(g(x))[/math]. Use the "Clear" button to erase the trace and turn it off. [/*][*]If you need to zoom out or scroll and adjust the domain of the inside function, you can click and drag the points labeled [math]a[/math] and [math]b[/math]. [/*][/list]

If we think of a function [math]f(x)=y[/math] as taking an input of [math]x[/math] and producing an out of [math]y[/math], then we can follow up one function with another function, creating a chain of functions, where the output of one function is used as the input of the next function. (It's like when you do a calculation on your calculator and use a previously stored value to do a new calculation.) Here is a chain:[br][br][math]x\to g(x)\to f(g(x))[/math][br][br]In this example [math]x[/math] is the input to the function [math]g[/math], which results in an output [math]g(x)[/math]. Then, [math]g(x)[/math] is used as the input of f to obtain [math]f(g(x))[/math] as the output of the composite function. Notice the order matters:[br][br][math]x\to f(x)\to g(f(x))[/math][br][br]In this chain, [math]x[/math] is the input of [math]f[/math], which produces the output [math]f(x)[/math]. Then, [math]f(x)[/math] is the input of the function [math]g[/math], which produces the output [math]g(f(x))[/math].