MATH 1113 PreCalculus

Day #8, The Algebra of Functions,
Part 2 (Chapter 5.1 in the textbook)

The Algebra of Functions, II:

Function Compositions:  There are many non-pointwise methods of combining functions to produce new functions.  If you take a course in differential equations, you will learn about the "convolution of functions," for example (If you'd like to explore them now, there's an amazing website from Johns Hopkins University: http://www.jhu.edu/~signals/index.html.  Click on the "Interactive Lecture Module."  It may make more sense after you've studied trigonometry, but it's fascinating, anyway.)

The first, and most important non-pointwise combination (and the only one we'll look at in this class), is called the "composition of two functions."  Our textbook, and most American sources, defines the composition:

Warning:  While this definition is standard, it is far from universal.  It uses what is called "prefix" notation -- read this as "f of g of x," or "f applied to g applied to x."  Many other sources -- especially in computer science applications -- define the composition in the opposite manner ("postfix" notation), and, as we'll see, it does make a difference!  Fortunately, postfix notation is usually written without the open circle:  f(g(x)) would be written (x)gf (read this as: "take x, then do "g," then do "f.")  See this Wikipedia article for more information (scroll down to "notations.")

Using the "input-output" approach to functions, to form the composition, for each input, x, we evaluate g(x), and the output of the g(x) function becomes the input in the "f" function.  So the graph of the resulting function will have the points (x, f(g(x)):

To make this idea a bit less complicated (looking), we notice that taking the value g(x) out on the y-axis, and then re-inserting it on the x-axis can also be accomplished by reflecting our arrow off the line y = x:

Tracing out all of these points gives the composition: f(g(x)):

Here is a video demonstrating the composition of two functions. 

DivX

QuickTime

I suggest that you stop it somewhere in the middle and slide it back and forth to see how the composition is created.  Also, I just noticed that the composition in the movie is g(f(x)), while in the discussion above and below, we've been looking at f(g(x)). 

Or, if you'd like an interactive approach, here's a new GeoGebra applet that creates (again), g(f(x)).

Applet

(Example #9, p. 371 gives another look at forming compositions graphically.)

The domain of f(g(x)) is the intersection of the domain of g(x) with the set of all x such that g(x) is in the domain of f:

, since g(x) is not defined for x < 0.

In Calculus, you will need to be able to decompose a "composite" function into its constituent parts; this decomposition is not unique, but you will learn a "best" decomposition that allows you to find the derivative of a composite function by using something called the "chain rule."  See the book (section 5.1) for practice algebraically forming compositions and decompositions.

Composition is non-commutative:

Most operations involving real numbers are either commutative, like addition and multiplication:

a + b = b + a and ab = ba, or else anti-commutative, like subtraction and division:

a - b = - (b - a) and a/b = 1/(b/a).

Function composition is neither.  Simply put, composing two functions in opposite order can produce very different, and non-comparable results.

For example, if f(x) = x² and g(x) = x + 1, we have that:

f(g(x)) = (x+1)² = x² + 2x + 1, while g(f(x)) = x² + 1.  This is why it is important to know how a particular source defines composition.

Page Last Modified: 21 August, 2008

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