This is a illustration of composite functions. A composite function [math]f \circ g = f(\;g(x)\;)[/math] takes the output of one function, [math]g[/math] and uses that as the input to another function [math]f[/math]. You can choose the functions from the drop down list. With the box checked both functions are shown as well as the flow of information. Moving the brown circle changes the [math]x[/math]. value. Follow the arrows to see how the output from the inside function is moved back to the abscissa ( [math]x[/math]) and then through the outside function to get the final result. Un-checking the box will show a single function made by combining the functions. [br][br]NOTE: there is a bug in the online version which does not plot the composite function correctly. Noting where the brown and red data flow arrows turn shows where the composite function curve should be located.

Note the domains and ranges of the various composite functions in relation to each of the other functions.[br]Try inverse functions like [math]sin(x) [/math] and [math]\text{asin}(x)[/math] or [math]e^x[/math] and ln(x).[br]What happens when you switch the order of [math]\text{asin}(x)[/math] and [math]sin(x) [/math] ?[br]When you change the check box does the result change?