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

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

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

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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