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Int Algebra: Functions and the Vertical Line Test


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About this Lesson

  • Type: Video Tutorial
  • Length: 7:07
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 76 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: Function Basics (3 lessons, $4.95)

A function is basically a machine that takes an input value (x) and processes it to produce an output value (y). With a function, if an x value is known, you can find the y value. When graphed, a curve is a function if it passes the vertical line test. In this lesson, Professor Burger will show you how the vertical line test means and how to recognize when a curve does not pass the vertical line test. The vertical line test looks to verify that, for every value of x, only one y value is produced. If something doesn't pass the vertical line test, it is called a relation and not a function.

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 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 UT-Austin 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, p-adic 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

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Founded in 1997, Thinkwell has succeeded in creating "next-generation" 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 technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


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Relations and Functions
An Introduction to Functions
Functions and the Vertical Line Test Page [1 of 2]
Okay, so now I want to start talking about a realm of mathematical objects that will follow us, not only through this
class, but any other math class we take, and that is the idea of a function. So what is a function? A function is really
just a machine where you sort of input something and then output one particular thing. So you really think about
input--something happens, and the output. Let me show you a very specific example so we can start to dig right in.
This is a function of x and it’s y = x2 + 10x + 25. Now, what does it mean to be a function? What it means is all you’ve
got to tell me to find y is what x equals. If you tell me x, then I can go off in a corner, like a little machine, and actually
figure out what y is. And what would I do? If you gave me the x I would just take that value and square it, add it to 10
times the value, and add 25, and that’s exactly y. Notice that y only depends upon x, and once you give me x I can
find that one y that’s going to actually satisfy this. So you really should think about it like a machine. And, in fact, let
me actually show you a function type machine.
So, for example, the way it works is you put something in and something else comes out. So let’s say that I want to
take a look at this function and see what happens when I plug in x = 1. So when I plug in x = 1, what would this
equal? x = 1, then what is y? Wherever I see an x I put in 1. So 12 + 10 x 1 + 25. And what does that equal? Well,
that equals 1 + 1 + 25, which equals 36. So y = 36 when x = 1. So the idea is when you input 1 into this function, out
should come 36. Let’s try it and see if that’s right. So the way this works is--you take 1 and you put it in, and then
what happens when I turn it? It goes through the function and out comes 36. See how that works. Isn’t that neat? So
that’s exactly what’s happening here. You put something in and something else comes out. Let’s try another example.
What if I were to put in x = 2? What would I see? Well, y would equal 22 + 10 x 2 + 25. Well, that would be 4 + 20 +
25, and that would equal 49. So if you input x = 2, output would be y = 49. Let’s try that and see if this works. So I
take a 2 and I’m going to input it into here, into the function press, and let’s see what happens. Watch it really
carefully. There it comes, right out. And as it comes out, what do we see? Voila! We see 49, just like we expected.
Now, in fact, you can do this with anything. For example, suppose you take something like this. What if you let x be
$5? It doesn’t make a difference what you plug in. What would you get out? I’d have $52 which is $25, and I’ve have
5 x 10 which is $50 + $25, so that would equal $100. So if this is really correct, I should be able to take a $5 bill and
put it in, and let’s see what happens. Oh-oh. You see it? This is looking cool, folks. Holy cow! Five dollars in and a
crisp, nice, beautiful $100 comes out. Folks, there’s nothing like the smell of money. This illustrates a very important
principle--math pays.
Anyway, you can now sort of see how these functions look and how they behave. Now, in fact, there’s a way to
visualize this thing. If you want to visualize it graphically, what does a function look like? A curve is a function if it
actually passes the vertical line test, and the vertical line test is just the following. If I put down a vertical line the curve
should hit the function at most at one place, and here’s why. What is means is for any x value you give me, there will
be at most one y value that will be associated with that x value. So, what that means is if I plug in a number, I get one
answer. Look at the following curve. This curve, which I’m showing you right here is actually not a function, and why
is it not a function? Because it fails the vertical line test. There are places on this curve where if I draw a vertical line,
you see, it actually crosses the curve in more than one place. And what does that mean in terms of our machine? It
means that if I insert that x value there’s not one y value that spits out. In fact, there would be three y values--one
over here, one over here, and one over here. So this is not a function, because one x value doesn’t give rise to a
unique y value. It’s only when for every x value we spit out one y value do we actually have a function. It’s a
machine. This machine--we get very confused--even over here you can see it would be very confusing. If I put in that
x value, where would the function spit out? Would it be that value there? Would it be this value? Or would it be this
value where the curve crosses the vertical line? So this is not a function. A function is something where no matter
what x value you give me, if it’s allowable, there will only be one answer, one output. So this is called the vertical line
test. You lay your vertical line down, and if the vertical line only touches at most one point, then you know this is a
function. If, in fact, there are places where the vertical line actually touches at more than one point, then you know it’s
not a function, it’s just called a relation, because it’s not a function because you give me any x value, and in fact, there
are more than one y value possibilities. So what does the machine spit out? It’s confused; it stops; it halts.
So a function is an object where you input one thing and one thing come out. If many things want to come out that’s
not a function, that’s merely a relation. So anyway, that’s the notion of a function, and now, believe it or not, we’re
Relations and Functions
An Introduction to Functions
Functions and the Vertical Line Test Page [2 of 2]
going to do variations on the theme of functions for as long as you can stand it, and I hope that’s pretty long because
there’s a lot of stuff I want to tell you. So functions are great. We’ll see graphs of them; equations of them. I’ll see you

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