Algebra Basics: What Are Functions? - Math Antics

0:06 Hi, I’m Rob.

0:07 Welcome to Math Antics!

0:08 In this algebra basics lesson, we’re gonna learn about functions.

0:12 Outside of the realm of math,

0:14 the word “function” simply refers to what something does.

0:16 But in math, the word “function” has a more specific meaning.

0:20 In math, a function is basically something that relates

0:23 or connects one “set” to another “set” in a particular way.

0:27 A set is just a group or collection of things.

0:30 Often it’s a collections of numbers, but it doesn’t have to be.

0:33 A set could be a collection of other things like letters,

0:37 names, or just about anything.

0:38 Sets are sometimes shown visually like this, but more often

0:42 you’ll see sets written using a common math notation where

0:44 some or all of the members of the set are

0:46 put inside curly brackets with commas between them like this.

0:51 A set can have a finite or an infinite number of elements.

0:54 For example, a set containing all the letters

0:56 of the alphabet has only 26 elements,

0:59 while the set of all integers has an infinite number of elements.

1:03 Okay, so a set is just a collection of things,

1:05 and a function relates one set to another.

1:08 But how exactly does it do that?

1:10 Well, to understand how functions work it will help if we start

1:13 by naming the two sets the input set and the output set.

1:16 A function is something that takes each value from an input set

1:20 and relates it (or maps it) to a value in an output set.

1:23 And you’ll often hear these input and output

1:25 sets referred to by special math names.

1:27 The input set is usually called “The Domain”

1:30 and the output set is usually called “The Range”.

1:33 And it’s really common to see some or all of a function’s

1:36 inputs and outputs listed in what we call a “function table”.

1:40 A function table normally has two columns: one on the left for the input values

1:44 and one on the right for the corresponding output values.

1:47 The function itself is often written above the function table

1:50 and in the form of some sort of mathematical rule or procedure.

1:54 For example, let’s say that the input set of a function

1:57 is a list of common polygon names like {triangle, square, pentagon,

2:01 hexagon and octagon} The function itself could be a simple rule that says,

2:06 “Output the number of sides.” That means,

2:08 if we input “triangle” into the function, the output will be 3.

2:12 And if we input “square” the output will be 4.

2:15 If we input “pentagon”, the output will be 5,

2:18 and so on… So this function simply relates

2:20 the name of a polygon to its number of sides.

2:23 That’s cool… but most of the functions that you’ll encounter

2:26 in Algebra will be a little more abstract than that.

2:29 They’ll usually just relate one variable to another variable

2:32 in the form of an equation …like this one: y= 2x In this equation,

2:37 if we treat ‘x’ as the set of numbers that we can input (the domain),

2:41 and ‘y’ as the set of numbers that we get as outputs (the range),

2:45 what we have is a very simple algebraic function.

2:48 And just like the polygon example, we can make a function table

2:52 to show some of the possible input-output combinations.

2:55 For this function, we could choose any number at all for the value of ‘x’,

2:58 but to keep things simple, let’s just try inputting 1, 2,

3:01 and 3 as values of ‘x’ and see what outputs we get for our table.

3:05 If we input the value 1 (in other words,

3:07 if we substitute the value 1 for the ‘x’ in our equation)

3:10 then we get y= 2× 1 which simplifies to y= 2.

3:15 And since ‘y’ is our output variable, we put a 2 in the output column.

3:20 Next, if we input the value 2 into our function,

3:23 we get y= 2× 2, which means y= 4.

3:26 So the output value is 4.

3:28 And last, if we input the value 3 into our function,

3:31 we get y= 2× 3, which means y= 6.

3:35 So the output value is 6.

3:37 …see the pattern?

3:38 For each input value, the output value is twice as big.

3:41 Which is what we would expect because the original equation says

3:45 that 'y' (the output) is equal to 2 times ‘x’ (the input) Okay,

3:49 so we’ve seen some examples of functions that relate inputs to outputs,

3:53 but there’s an important limitation about functions that we need to know.

3:56 To understand what that limitation is,

3:58 let’s try to make a function table for the equation ‘y squared’ equals ‘x’.

4:02 Again, the ‘x’ variable in this equation will be our set

4:05 of inputs and the ‘y’ variable will be our set of outputs.

4:08 Since ‘y’ is our output variable,

4:10 it will help if we first solve this equation for ‘y’

4:13 and we do that by taking the square root of both sides.

4:16 But because of negative numbers,

4:17 we need to take both the positive and negative root

4:20 of ‘x’ since there are two possible solutions to our equation.

4:23 But, won’t that mess up our function table?

4:25 If we input an ‘x’ value of 4, the positive (or principal) root would be 2,

4:30 but we also have the negative root as a solution.

4:32 If x= 4, then y= 2 and y= -2 are

4:37 BOTH possible solutions to the equation ‘y squared’ equals ‘x’.

4:41 So in this case, for each value of ‘x’ that we input into the equation,

4:44 we’ll get TWO values of ‘y’ as outputs.

4:46 Can a function do that?

4:48 [whistle blowing] Upon review, the equation gave two outputs for a single input,

4:53 therefore it’s ruled not a function.

4:56 You see, functions aren’t allowed to have what we call “one-to-many” relations,

5:00 where one particular input value could result in many different output values.

5:04 “One-to-many” relations certainly do exist as we can see from this example,

5:08 but we don’t call them functions.

5:10 For something to be called a function,

5:12 it has to produce only one output value for each input value.

5:16 So a function doesn’t just relates a set of inputs to a set of outputs.

5:20 A function relates a member of an input

5:22 set to exactly one member of an output set.

5:25 The equation y= 2x qualifies as function because no matter what number

5:29 you put in, you’ll always get just one number as an output.

5:33 But the equation ‘y squared’ equals ‘x’ does not qualify

5:36 as a function because a single input can produce more than one output.

5:40 Let’s look at another simple algebraic equation

5:42 to see if it’s a function: y= x+ 1.

5:46 Again, the ‘x’ values will be inputs (the domain)

5:49 and the ‘y’ values will be the outputs (the range).

5:52 Let’s quickly generate a function table for a few possible input values,

5:56 like the integers -3 through +3.

5:58 If you watched our last video about graphing on the coordinate plane,

6:01 you may notice that each row of this function

6:04 table is basically just an ordered pair.

6:06 It’s an ‘x’ value followed by a ‘y’ value.

6:09 We could even re-write all the inputs and outputs

6:11 in ordered pair form if we wanted to.

6:13 And that means, you can also GRAPH all

6:15 of these pairs of inputs and outputs on the coordinate plane.

6:18 You can GRAPH a function!

6:20 Here are the points from our function table plotted on the coordinate plane,

6:24 and here’s the resulting graph we get if we connect those points.

6:27 It forms a straight line and it’s

6:29 an example of what is called a “linear function”.

6:32 In Algebra, there are lots of different

6:34 kinds of functions that have interesting graphs:

6:36 quadratic functions, cubic functions, trig functions, and many more.

6:41 These graphs may look like just a bunch of squiggly lines,

6:44 but they’re all functions.

6:45 And we tell they’re functions just by looking

6:47 at their graphs because they all pass the “Vertical Line Test”.

6:51 Remember how functions aren’t allowed to have more

6:54 than one output value for a particular input value?

6:56 Well, the Vertical Line Test helps us see if a graph has

6:59 any of those one-to-many relations that would disqualify it as a function.

7:03 Here’s how it works… Imagine that a vertical line is drawn

7:07 on the same coordinate plane as the graph that you want to test.

7:10 Then, imagine moving that vertical line left and right across the domain,

7:14 paying close attention to the point where

7:17 the vertical line intersects with the graph.

7:20 If that vertical line only intersects the graph at exactly

7:23 one point for every possible value of ‘x’ in the domain,

7:27 then that means there’s only one output value for each input value.

7:31 There’s only one ‘y’ value for each ‘x’

7:33 value so the graph qualifies as a function.

7:36 Okay, so all of these graphs pass the Vertical Line Test and are functions.

7:40 But what’s an example of a graph that doesn’t pass the Vertical Line Test?

7:44 Well here’s one.

7:45 It’s the graph of our equation ‘y squared’ equals ‘x’.

7:49 The domain of this equation doesn’t include any negative input values,

7:52 so there are some places where our vertical

7:54 line wouldn’t interest the graph at all.

7:56 And that’s okay.

7:57 And there’s one place where the vertical line would intersect

8:00 the graph at just one point, which is also okay.

8:03 But, as we move to the right on the ‘x’ axis,

8:06 you can see that our vertical line is now intersecting the curve in TWO places.

8:10 That means this equation is giving us

8:12 two possible outputs for some of its inputs,

8:14 which means that it’s not considered a function.

8:17 Okay… now before we wrap up,

8:19 we need to talk briefly about some common function notation that can

8:22 be pretty confusing the first time you see it in math books.

8:26 So far, we’ve been writing functions like this: y= 2x and y= x+

8:31 1 But you’ll often see these same exact functions written like this instead.

8:36 But Why?

8:37 Why did the variable ‘y’ get replaced by that ‘f’ parentheses ‘x’ thingy?

8:42 And what does that even mean?

8:44 Well, it turns out that a really common way to represent a function is

8:47 this… This notation simply means that a function

8:50 (named ‘f’) takes an input value

8:52 (named ‘x’) and gives an output value (named ‘y’) And you say it like

8:56 this: A function of ‘x’ equals ‘y’ or ‘f’ of ‘x’ equals ‘y’ for short.

9:02 The problem with this notation is that you

9:04 could easily misinterpret it as a variable

9:06 ‘f’ being multiplied implicitly by a variable ‘x’ to give an answer of ‘y’.

9:11 But that’s NOT what this means.

9:13 In this case, ’f’ is not the name of a variable and it’s not being multiplied.

9:17 Instead, ’f’ is the name of the function.

9:20 It would be a lot more clear

9:21 if mathematicians just used the entire word “function”

9:23 as the name and then used the names “input” and “output” instead of ‘x’ and ‘y’.

9:28 These two notations mean exactly the same thing.

9:31 But the first one uses an abbreviation for the function

9:34 name and standard variable names for the input and output.

9:37 These are the most common names, but you could use others if you wanted to.

9:41 Okay, so that’s the basic notation,

9:43 but how did the equation get changed to f(x) instead of ‘y’?

9:48 Well, it comes from the idea that if two things are equal in math,

9:52 you can substitute one thing for the other.

9:54 Since we’ve agreed on this general notation for a function,

9:57 f(x)= y, that means you can use f(x) or ‘y’ interchangeable.

10:02 Either one can represent the output set of a function.

10:06 But if they’re interchangeable,

10:07 why would you use the more complicated f(x) when you could just use ‘y’ instead?

10:12 Well, using f(x) highlights the fact that you’re dealing with a function

10:16 with a specific input variable and not just an equation.

10:20 And… it gives us a handy notation for evaluating functions for specific values.

10:24 For example, you could start off by saying, let the function f(x)= 3x+ 2.

10:30 Then you could then ask someone to evaluate the function

10:34 for the input value 4 by saying what is f(4).

10:37 That means you’ll substitute a 4 in place of any ‘x’s that are in the function.

10:42 For this function, that would mean f(4)= 14.

10:46 And you could do this for other values too.

10:48 f(5)= 17, and f(6)= 20 Pretty easy, huh?

10:53 Alright, so that’s what functions are in math.

10:56 They’re things that relate an input value to exactly one output value.

11:00 And the set of all input values is called the domain

11:02 while the set of output values is usually called the range.

11:06 In algebra, functions typically come in the form

11:08 of equations that can be graphed on the coordinate plane by treating

11:12 the input and output values as ordered pairs.

11:15 Of course, there’s a LOT more to learn about functions,

11:18 but this basic introduction should help you

11:20 get started working with them in Algebra.

11:22 Don’t forget to practice using what you’ve

11:24 learned in this video by doing some exercises.

11:26 As always, thanks for watching Math Antics and I’ll see ya next time.

11:31 Learn more at www.mathantics.com