Graphical, algebraic, and numerical demonstrations of the composition , also written as .

Here we have two function graphs, and . Our input to is . Drag the blue point along 's -axis to change its value. As you do this, the value of changes. This output value, , then becomes the input value to , so that the value of is given by the composition .
You can change the definitions of and by typing new function expressions into their respective input boxes. Also note that you can drag the coordinate planes as well as the input boxes and the text boxes so that you can see the graphs, should you decide to change them. Click the circle arrows at the top right of the graph of to reset the display.
Algebraically, the input function is "plugged in" to the function to form the output, the composite function . The resulting function is shown in the text box on the graph. Note that the expression shown for is simplified, so it might look different than you expect depending on the functions you choose.
Graphically, you can see how the composition works by moving the -value on the graph. The -value output from for this -value is then plotted as the input value to on the -axis of the graph. The output of using this input gives us the final value of the composition.
Numerically, the tables work similarly to the graphical action. A set of two tables is shown. In the table on the graph, is limited to a set of selected values, from to in steps of . The output of for each of these inputs is given. These output values become the input values to the table shown on the graph. The output of at these values is the value of the composition of the two functions for the -value you selected by dragging the point.