Composition of Functions

Author:Ken Schwartz

Topic:Functions

Graphical, algebraic, and numerical demonstrations of the composition

, also written as

Here we have two function graphs,

and

. Our input to

. Drag the blue

point along

-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

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.

Composition of Functions

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