Algebra Basics: What Are Functions? - Math Antics - Free Educational videos for Students in K-12

Algebra Basics: What Are Functions? - Math Antics - By Mathantics

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00:03	Oh hi , I'm rob . Welcome to Math Antics
00:08	. In this algebra basics lesson , we're going to
00:10	learn about functions outside of the realm of math .
00:13	The word function simply refers to what something does ,
00:16	but in math the word function has more specific meaning
00:19	. In math , a function is basically something that
00:22	relates or connects one set to another set in a
00:25	particular way a set is just a group or collection
00:29	of things . Often it's a collection of numbers but
00:31	it doesn't have to be . A set can be
00:33	a collection of other things like letters , names or
00:36	just about anything . Sets are sometimes shown visually like
00:40	this , but more often you'll see sets written using
00:42	a common math notation where some or all of the
00:45	members of the set are put inside curly brackets with
00:48	commas between them like this , A set can have
00:51	a finite or infinite number of elements . For example
00:54	, a set containing all the letters of the Alphabet
00:56	has only 26 elements , while a set of all
00:59	integers has an infinite number of elements . Okay ,
01:03	so a set is just a collection of things and
01:05	a function relates one set to another . But how
01:08	exactly does it do that ? Well to understand how
01:11	functions work ? It will help if we start by
01:12	naming the two sets , the input set and the
01:14	output set . A function is something that takes each
01:18	value from an input set and relates it or maps
01:20	it to evaluate an output set and you'll often hear
01:23	these input and output sets referred to by special math
01:26	names . The input set is usually called the domain
01:29	and the output set is usually called the range and
01:33	it's really common to see some or all of the
01:35	functions inputs and outputs listed in what we call a
01:38	function table . A function table normally has two columns
01:42	, one on the left for the input values and
01:44	one on the right for the corresponding output values .
01:47	The function itself is often written above the function table
01:50	and in the form of some sort of mathematical rule
01:52	or procedure , for example , let's say that the
01:55	input set of a function is a list of common
01:57	polygon names like triangle , square , pentagon , hexagon
02:01	and octagon . The function itself could be a simple
02:04	rule that says output the number of sites . That
02:07	means if we input triangle into the function , the
02:10	output will be three . And if we can put
02:12	square , the output will be four . If we
02:14	can put pentagon , the output will be five and
02:17	so on . So this function simply relates the name
02:20	of a polygon to its number of sides . That's
02:23	cool . But most of the functions that you'll encounter
02:25	in algebra will be a little more abstract than that
02:28	. They'll usually just relate one variable to another variable
02:31	in the form of an equation like this one ,
02:33	Y equals two X . In this equation , if
02:37	we treat X as a set of numbers that we
02:39	can input the domain and why as the set of
02:42	numbers that we get as outputs the range . What
02:44	we have is a very simple algebraic function . And
02:48	just like the polygon example , we can make a
02:50	function table to show some of the possible input output
02:53	combinations for this function , we could choose any number
02:56	at all for the value of X . But to
02:58	keep things simple , let's just try and putting 12
03:00	and three as values of X and see what outputs
03:02	we get for our table . If we can put
03:05	the value one , in other words , if we
03:07	substitute the value one for the X and our equation
03:10	then we get y equals two times one , which
03:13	simplifies to Y equals two . And since why is
03:16	our output variable we put it to in the output
03:18	column next . If we put the value to into
03:21	our function we get y equals two times two ,
03:24	which means why equals four . So the output value
03:27	is four and last . If we have put the
03:29	value three into our function we get y equals two
03:32	times three , which means y equals six . So
03:34	the output value is six . See the pattern for
03:38	each input value . The output value is twice as
03:40	big , which is what we would expect because the
03:42	original equation says that why the output is equal to
03:46	two times X . The input . Okay . So
03:49	we've seen some examples of functions that relate inputs to
03:51	outputs but there's an important limitation about functions that we
03:54	need to know to understand what that limitation is .
03:57	Let's try to make a function table for the equation
03:59	Y squared equals X . Again , the X variable
04:03	in this equation will be our set of inputs and
04:05	the Y variable will be our set of outputs .
04:08	Since why is our output variable ? It will help
04:10	if we first solve this equation for why and we
04:12	do that by taking the square root of both sides
04:15	. But because of negative numbers , we need to
04:17	take both the positive and negative root of X .
04:20	Since there are two possible solutions to the equation .
04:23	But won't that mess up our function table if we
04:25	have put an X value of four , the positive
04:27	or principal route would be to but we also have
04:30	the negative route as a solution . If x equals
04:32	four , then y equals two and Y equals negative
04:35	two are both possible solutions to the equation Y squared
04:39	equals X . So in this case for each value
04:42	of X that we have put into the equation will
04:44	get to values of Y as outputs can a function
04:46	do that ? You see functions aren't allowed to have
04:57	what we call one Too many relations where one particular
05:00	input value could result in many different output values .
05:04	One too many relations certainly do exist . As we
05:07	can see from this example , but we don't call
05:09	them functions for something to be called . A function
05:11	, it has to produce only one output value for
05:14	each input value . So a function doesn't just relate
05:17	a set of inputs to a set of outputs .
05:19	A function relates a member of an input set to
05:22	exactly one member of an output set . The equation
05:26	y equals two . X qualifies as a function because
05:28	no matter what number you put in , you always
05:30	get just one number as an output . But the
05:33	equation why squared equals X . Does not qualify as
05:36	a function because a single input can produce more than
05:38	one output . Let's look at another simple algebraic equation
05:42	to see if it's a function Y equals X plus
05:45	one . Again . The X values will be inputs
05:48	the domain and the Y values will be the outputs
05:50	. The range . Let's quickly generate a function table
05:53	for a few possible input values like the integers negative
05:56	three through positive three . If you watched our last
05:59	video about graphing on the coordinate plain , you may
06:01	notice that each roll of this function table is basically
06:04	just an ordered pair . It's an X . Value
06:07	followed by a Y value . We could even rewrite
06:10	all the inputs and outputs in order to perform if
06:12	we wanted to . And that means you can also
06:14	graph all of these pairs of inputs and outputs on
06:17	the coordinate plain . You can graph a function here
06:20	are the points from our function table plotted on the
06:22	coordinate plain . And here's the resulting graph we get
06:25	if we connect those points , it forms a straight
06:28	line and it's an example of what is called a
06:30	linear function in algebra . There are lots of different
06:33	kinds of functions that have interesting graphs , Quadratic functions
06:37	, cubic functions , trig functions and many more .
06:40	These graphs may look like just a bunch of squiggly
06:42	lines , but they're all functions and we can tell
06:45	their functions just by looking at their graphs because they
06:48	all pass the vertical line test . Remember how functions
06:52	aren't allowed to have more than one output value for
06:54	a particular input value ? Well , the vertical line
06:57	test helps us see if a graph has any of
06:59	those one . Too many relations that would disqualify it
07:02	as a function . Here's how it works . Imagine
07:05	that a vertical line is drawn on the same coordinate
07:08	plane as the graph that you want to test .
07:10	Then imagine moving that vertical line left and right across
07:13	the domain , paying close attention to the point where
07:16	the vertical line intersects with the graph . If that
07:20	vertical line only intersects the graph at exactly one point
07:24	for every possible value of X in the domain ,
07:26	then that means that there's only one output value for
07:29	each input value . There's only one why value for
07:32	each X . Value . So the graph qualifies as
07:34	a function . Okay , So all of these graphs
07:38	pass the vertical line test and our functions . But
07:40	what's an example of a graph that doesn't pass the
07:42	vertical line test ? Well , here's one , it's
07:45	the graph of our equation , Y squared equals X
07:49	. The domain of this equation doesn't include any negative
07:51	input values . So there's some places where a vertical
07:54	line wouldn't intersect the graph at all . And that's
07:56	okay . And there is one place where the vertical
07:58	line would intersect the graph at just one point which
08:01	is also okay . But as we move to the
08:04	right on the X axis , you can see that
08:06	are vertical line is now intersecting the curve in two
08:09	places . That means this equation is given us two
08:12	possible outputs for some of its inputs , which means
08:14	that it's not considered function . Okay . Now ,
08:18	before we wrap up , we need to talk briefly
08:20	about some common function notation . That can be pretty
08:22	confusing . The first time you see it in math
08:24	books so far , we've been writing functions like this
08:28	. Y equals two X and Y equals X plus
08:31	one . But you'll often see these exact same functions
08:34	written like this instead . But why Why did the
08:37	variable why get replaced with that F . Parentheses X
08:40	. Thing . And what does that even mean ?
08:43	Well , it turns out that a really common way
08:45	to represent a function . Is this ? This notation
08:48	simply means that a function named F takes an input
08:51	value named X and gives an output value named Why
08:55	? And you say it like this , A function
08:57	of X equals Y , or F of X equals
09:00	Y . For short . The problem with this notation
09:03	is that you could easily misinterpret it as a variable
09:06	F being multiplied implicitly by a variable X . To
09:09	give an answer of why . But that's not what
09:11	this means . In this case it is not the
09:14	name of the variable and it's not being multiplied instead
09:17	, F is the name of a function . It
09:19	would be a lot more clear if mathematicians just use
09:22	the entire word function as the name and then use
09:24	the names input and output instead of X and Y
09:27	. These two notations mean exactly the same thing ,
09:30	but the first one uses an abbreviation for the function
09:33	name and standard variable names for the input and output
09:36	. These are the most common names , but you
09:38	could use others if you wanted to . Okay ,
09:41	so that's the basic notation . But how did the
09:44	equation get changed to F of X instead of Y
09:47	? Well , it comes from the idea that if
09:49	two things are equal in math , you can substitute
09:52	one thing for the other . Since we've agreed on
09:54	this general notation for a function F of X equals
09:57	Y . That means that you can use F of
10:00	X or Y interchangeably . Either one can represent the
10:03	output set of a function . But if they're interchangeable
10:07	, why would you use the more complicated ffx when
10:10	you could just use Y instead ? Well , using
10:13	F of X highlights the fact that you're dealing with
10:15	a function with a specific input variable and not just
10:18	an equation . And it gives us a handy notation
10:21	for evaluating functions for specific values . For example ,
10:25	you can start off by saying let the function F
10:27	of X equals three X plus two . Then you
10:30	could ask someone to evaluate the function for the input
10:33	value four by saying what is F of four ?
10:36	That means you'll substitute of four in place of any
10:39	exes that are in the function For this function .
10:42	That would mean f of four equals 14 and you
10:45	can do this for other values to f F five
10:48	equals 17 and ff six equals 20 . Pretty easy
10:51	. Huh ? All right . So that's what functions
10:54	are in math . There are things that relate an
10:56	import value to exactly one alpa value and the set
11:00	of all input values is called the domain where the
11:02	set of all output values is usually called The range
11:05	in algebra , functions typically come in the form of
11:08	equations that can be graphed on the coordinate plain by
11:11	treating the input and output values as ordered pairs .
11:14	Of course , there's a lot more to learn about
11:17	functions , but this basic introduction should help you get
11:19	started working with them in algebra . Don't forget to
11:22	practice using what you've learned in this video by doing
11:25	some exercises as always . Thanks for watching Math Antics
11:28	and I'll see you next time learn more at Math
11:31	Antics dot com .

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