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About this Lesson
 Type: Video Tutorial
 Length: 9:37
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 104 MB
 Posted: 12/02/2008
This lesson is part of the following series:
College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
College Algebra: Relations and Functions (57 lessons, $74.25)
Trigonometry: Algebra Prerequisites (60 lessons, $69.30)
Intermediate Algebra Review (25 lessons, $49.50)
College Algebra: Composite Functions (5 lessons, $7.92)
In this lesson, you will learn about a method that you can use to combine functions. The composition of two functions is the way to combine two functions. In this lesson, you will learn how to combine functions (for example, to find f(g(x)) ). There are specific ways to denote these types of composite functions, and you will also learn how to correctly write composite functions (f(g(x)) or (f o g)(x) ). To compose a function (find the composition of functions), you'll have to take the answer of one function and plug it into the other function (to find something like, 'g composed of f of 3'. Professor Burger will also highlight why g(f(3)) is not always equal to f(g(3)).
This lesson is perfect for review for a CLEP test, midterm, final, summer school, or personal growth!
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Intermediate Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/intermediatealgebra. The full course covers real numbers, equations and inequalities, exponents and polynomials, rational expressions, roots and radicals, relations and functions, the straight line, systems of equations, quadratic equations and quadratic inequalities, conic sections, inverse and exponential and logarithmic functions, and a variety of other AP algebra and advanced algebra.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
He has also taught at UTAustin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, padic analysis, the geometry of numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
 Thinkwell
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 Joined:
11/14/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
More..Recent Reviews
Relations and Functions
The Algebra of Functions
Composite Functions Page [1 of 2]
Okay, now I want to tell you about this brand new way of combining functions that just doesn’t exist with respect to
numbers. So how would this look? Well, let’s think about it. Suppose I have two functions. One of them, let’s say
f(x) = 3x  1, and suppose I have another one, I’ll call it g(x) and that = 2x2 + x + 1. So these are two functions, and
remember what these things mean. They mean that if you give me an x I can plug it in here and find out what f(x) is; I
can give you the y value.
Well, we talked about adding, subtracting, multiplying and dividing these functions. But what about actually putting
these two functions together and actually composing them into one function in the following sense? Suppose that I
first take an x value and plug it in here, and actually get some sort of output, but then use that output to actually put
into g? So I sort of take f and sort of shove it right into g. So I sort of not just add them or subtract, them I sort of
actually inject f into g. Let me say again how that would work. I would take an x and I would evaluate this, and then I
would take that as the answer and put that in for x here. So visually, it might look like this. You have the x’s here, and
that will then take via f, values to y’s.
But what if I take those y values and use them now as inputs for g? Then I’ll get something else out. How would this
look? Well, let me show you with this example. If you take a look at the f function, here I plot some points. For
example, at 1 if you plug that in, we get something weird herewe get 4 here. You plug in zero, you get 1. If we
plug in 1, we get 2, and 2 we get 5. I think there’s a typo here, let’s hope. So anyway, there’s a chart of some values
for f.
Let me show you a chart of some values for g. For example, if I put in 1 for x, what do I see? I see a 2  1 + 1 is 2. If
I put in a zero for x here, I see 0, 0, 1. If I put in a 1 I see 2 and 1 and 1 is 4. If I put in a 2 I see 4 times 2 is 8, 9, 10,
11. So, fact, this is a chart for the g values, and this is a chart here for the f values. So to see this kind of thing in
action, let’s see what happens if, for example, I start with zero. Suppose I start with zero and plug it in here. If I plug
in zero that would give me a 1 as an answer. But what if I take that 1 and use it as input into g? So I go to here, 1,
it should spit out 2. So I’m basically using one function to get an answer, and then I take that answer and put it into the
next function. We can’t do that with numbers. Th ere’s no such thing as putting numbers into another number. But yet
here we can actually do it.
Let me show you how this would look with these function machines that we saw awhile back. So I have an f machine
and a g machine. The first thing I want to do is take the zero and put it through the f machine. What should happen?
Well, we should see the value of the function when I evaluate f at zero. So let’s try that right now. I put that in; let’s
see what happens. Out comesvoilawhat we saw on the chart, a 1. So, in fact, there’s the 1. Now, what I want to
do is take that answer and immediately put it into the g function. So I take that answer and run it into the g function,
what do I see? I see 2. So the final answer is 2. So I started with zero, that gave me 1, and I plugged 1 into g, and I
got 2. There’s the answer. And again, you can see that on the chart. I started with zero, that gave me a 1, and then
I took the 1 and put it into this function, and when I plugged away, I got the 2.
Let’s try another example. How about if I plugged in a 1? If I put in a 1 for f in x here, then that would give me a 2,
and what if I took that 2 and put that back into the g function for x? Then that should give me 11. You see how I take
the 1, that gives me a 2, and I take the 2 and plug that in, and that gives me an 11. Let’s see that principle in action.
So I’m back to here again. Now I take a 1, and if I take the 1 and put it right in, let’s see what happens. Out comes
well, there’s the 2 we expected, so that’s right here. That’s the 2, and now what I have to do is take the 2 and put that
through the g machine. So I take the answer and put that through the g machine, and let’s see if that gives us the 11
that we thought. It gives us the 11, just like we thought.
So, in fact, this is a way of taking a function and combining it with another function that’s not adding, subtracting,
multiplying, or dividing, but instead is composing the two functions together. I take an answer from one function and
use that as the input from the other. So I have an input to f, it comes out as output to go as input into the next
function. So this is called the composition of functions, and in fact, the way you would write this is exactly as you may
think you would say it. For example, let’s do the zero example once again. If I look at f(0), that’s some number. What
number is it? f(0) is 1. And now if I take that number and use it as input into the g function, what I would do is just
write g of all of this, g(f(0)). What does that mean? It means you first find f(0)you first find this quantity, and then
once you have that number, you plug that into the g function. This is the composition of functions. Sometimes this is
Relations and Functions
The Algebra of Functions
Composite Functions Page [2 of 2]
denoted using the following notation. You write the g function and then a circle, and then the f function. So it’s like
“gof.” It doesn’t mean multiplication. What it means is the composition of two functions. If you have an x value you
first see where x sends it to, and then take that answer and put that answer into g.
So let’s do some examples. For example, what would “gof”or g composed with f, as normal people read itof 3 be?
Well, what would I do? First I would figure out f(3), that’s this thing, and then I would see where g sends it to. So
what’s f(3)? Well, I go back up to the function f and I plug in a 3 wherever I see an x, and I would see 3 times 3 minus
1. Well, that’s 8. So this would be g(8). And so now what I have to do is go back to the g function and find out what
g(8) is. Well, what’s g(8)? Let’s see if you can do that really fast. So g(8) would equal 2 times 82, which is 64, plus 8,
plus 1, which is 9. So here we’d have 128 + 9, which equal 137. So this answer would be 137. So what is g
composed with f(3)? It would equal 137, and how did I get it? I first took the 3 and used it as input into the f function.
I got that answer and then took the answer and used it as input into the g function.
All right. Let’s try one last example, just to see that, in fact, this can actually go the other direction. What if I took the
following“fog”f composed with g(3)? What would that mean? That would mean first I figure out what g(3) is. I just
first do the g. Whatever it says here, I do that first, and then I compose that, take the answer, and put that into the f
function. So what’s g(3)? Well, if I put in 3 for g, what do I see? I’ll see a 9 times 2, that’s 18. 18 + 3 is what? 21,
and 1 is 22. So this is going to be f(22). So that answer is 22. Now I take that answer and I put it into the f function,
so I put 22 in for x, which would be what? 66  1, which is 65.
So notice that f composed with g(3) is 65, however, g composed with f(3) is 137. They’re not always the same,
because I’m plugging first into one, and then into the other. So it’s important to understand what the differences
between these two things are. This says, “First take the number 3 and plug it into g, and take that answer and plug it
into f.” Whereas this says, “Take the number 3 and first plug it into f, and then take that other answer and plug it into
g.” The composition of functions. We’ll take a look at a lot of examples of these coming up next to get a real feel for
how to compose function, a whole new way to actually combine functions together.
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I was struggling to grasp this concept when reviewing for an exam, as the examples in my text were not sufficient enough. This video was able to articulate the concept in a very clear manner and I can finally exhale and feel confident going into my exam. Thanks!