Hi, I’m Rob. Welcome to Math Antics! In this algebra basics lesson, we’re gonna learn about functions. Outside of the realm of math, the word “function” simply refers to what something does. But in math, the word “function” has a more specific meaning. In math, a function is basically something that relates or connects one “set” to another “set” in a particular way. A set is just a group or collection of things. Often it’s a collections of numbers, but it doesn’t have to be. A set could be a collection of other things like letters, names, or just about anything. Sets are sometimes shown visually like this, but more often you’ll see sets written using a common math notation where some or all of the members of the set are put inside curly brackets with commas between them like this. A set can have a finite or an infinite number of elements. For example, a set containing all the letters of the alphabet has only 26 elements, while the set of all integers has an infinite number of elements. Okay, so a set is just a collection of things, and a function relates one set to another. But how exactly does it do that? Well, to understand how functions work it will help if we start by naming the two sets the input set and the output set. A function is something that takes each value from an input set and relates it (or maps it) to a value in an output set. And you’ll often hear these input and output sets referred to by special math names. The input set is usually called “The Domain” and the output set is usually called “The Range”. And it’s really common to see some or all of a function’s inputs and outputs listed in what we call a “function table”. A function table normally has two columns: one on the left for the input values and one on the right for the corresponding output values. The function itself is often written above the function table and in the form of some sort of mathematical rule or procedure. For example, let’s say that the input set of a function is a list of common polygon names like {triangle, square, pentagon, hexagon and octagon} The function itself could be a simple rule that says, “Output the number of sides.” That means, if we input “triangle” into the function, the output will be 3. And if we input “square” the output will be 4. If we input “pentagon”, the output will be 5, and so on… So this function simply relates the name of a polygon to its number of sides. That’s cool… but most of the functions that you’ll encounter in Algebra will be a little more abstract than that. They’ll usually just relate one variable to another variable in the form of an equation …like this one: y=2x In this equation, if we treat ‘x’ as the set of numbers that we can input (the domain), and ‘y’ as the set of numbers that we get as outputs (the range), what we have is a very simple algebraic function. And just like the polygon example, we can make a function table to show some of the possible input-output combinations. For this function, we could choose any number at all for the value of ‘x’, but to keep things simple, let’s just try inputting 1, 2, and 3 as values of ‘x’ and see what outputs we get for our table. If we input the value 1 (in other words, if we substitute the value 1 for the ‘x’ in our equation) then we get y=2 × 1 which simplifies to y=2. And since ‘y’ is our output variable, we put a 2 in the output column. Next, if we input the value 2 into our function, we get y=2 × 2, which means y=4. So the output value is 4. And last, if we input the value 3 into our function, we get y=2 × 3, which means y=6. So the output value is 6. …see the pattern? For each input value, the output value is twice as big. Which is what we would expect because the original equation says that ‘y’ (the output) is equal to 2 times ‘x’ (the input) Okay, so we’ve seen some examples of functions that relate inputs to outputs, but there’s an important limitation about functions that we need to know. To understand what that limitation is, let’s try to make a function table for the equation ‘y squared’ equals ‘x’. Again, the ‘x’ variable in this equation will be our set of inputs and the ‘y’ variable will be our set of outputs. Since ‘y’ is our output variable, it will help if we first solve this equation for ‘y’ and we do that by taking the square root of both sides. But because of negative numbers, we need to take both the positive and negative root of ‘x’ since there are two possible solutions to our equation. But, won’t that mess up our function table? If we input an ‘x’ value of 4, the positive (or principal) root would be 2, but we also have the negative root as a solution. If x=4, then y=2 and y=-2 are BOTH possible solutions to the equation ‘y squared’ equals ‘x’. So in this case, for each value of ‘x’ that we input into the equation, we’ll get TWO values of ‘y’ as outputs. Can a function do that? [whistle blowing] Upon review, the equation gave two outputs for a single input, therefore it’s ruled not a function. You see, functions aren’t allowed to have what we call “one-to-many” relations, where one particular input value could result in many different output values. “One-to-many” relations certainly do exist as we can see from this example, but we don’t call them functions. For something to be called a function, it has to produce only one output value for each input value. So a function doesn’t just relates a set of inputs to a set of outputs. A function relates a member of an input set to exactly one member of an output set. The equation y=2x qualifies as function because no matter what number you put in, you’ll always get just one number as an output. But the equation ‘y squared’ equals ‘x’ does not qualify as a function because a single input can produce more than one output. Let’s look at another simple algebraic equation to see if it’s a function: y=x + 1. Again, the ‘x’ values will be inputs (the domain) and the ‘y’ values will be the outputs (the range). Let’s quickly generate a function table for a few possible input values, like the integers -3 through +3. If you watched our last video about graphing on the coordinate plane, you may notice that each row of this function table is basically just an ordered pair. It’s an ‘x’ value followed by a ‘y’ value. We could even re-write all the inputs and outputs in ordered pair form if we wanted to. And that means, you can also GRAPH all of these pairs of inputs and outputs on the coordinate plane. You can GRAPH a function! Here are the points from our function table plotted on the coordinate plane, and here’s the resulting graph we get if we connect those points. It forms a straight line and it’s an example of what is called a “linear function”. In Algebra, there are lots of different kinds of functions that have interesting graphs: quadratic functions, cubic functions, trig functions, and many more. These graphs may look like just a bunch of squiggly lines, but they’re all functions. And we tell they’re functions just by looking at their graphs because they all pass the “Vertical Line Test”. Remember how functions aren’t allowed to have more than one output value for a particular input value? Well, the Vertical Line Test helps us see if a graph has any of those one-to-many relations that would disqualify it as a function. Here’s how it works… Imagine that a vertical line is drawn on the same coordinate plane as the graph that you want to test. Then, imagine moving that vertical line left and right across the domain, paying close attention to the point where the vertical line intersects with the graph. If that vertical line only intersects the graph at exactly one point for every possible value of ‘x’ in the domain, then that means there’s only one output value for each input value. There’s only one ‘y’ value for each ‘x’ value so the graph qualifies as a function. Okay, so all of these graphs pass the Vertical Line Test and are functions. But what’s an example of a graph that doesn’t pass the Vertical Line Test? Well here’s one. It’s the graph of our equation ‘y squared’ equals ‘x’. The domain of this equation doesn’t include any negative input values, so there are some places where our vertical line wouldn’t interest the graph at all. And that’s okay. And there’s one place where the vertical line would intersect the graph at just one point, which is also okay. But, as we move to the right on the ‘x’ axis, you can see that our vertical line is now intersecting the curve in TWO places. That means this equation is giving us two possible outputs for some of its inputs, which means that it’s not considered a function. Okay… now before we wrap up, we need to talk briefly about some common function notation that can be pretty confusing the first time you see it in math books. So far, we’ve been writing functions like this: y=2x and y=x + 1 But you’ll often see these same exact functions written like this instead. But Why? Why did the variable ‘y’ get replaced by that ‘f’ parentheses ‘x’ thingy? And what does that even mean? Well, it turns out that a really common way to represent a function is this… This notation simply means that a function (named ‘f’) takes an input value (named ‘x’) and gives an output value (named ‘y’) And you say it like this: A function of ‘x’ equals ‘y’ or ‘f’ of ‘x’ equals ‘y’ for short. The problem with this notation is that you could easily misinterpret it as a variable ‘f’ being multiplied implicitly by a variable ‘x’ to give an answer of ‘y’. But that’s NOT what this means. In this case, ’f’ is not the name of a variable and it’s not being multiplied. Instead, ’f’ is the name of the function. It would be a lot more clear if mathematicians just used the entire word “function” as the name and then used the names “input” and “output” instead of ‘x’ and ‘y’. These two notations mean exactly the same thing. But the first one uses an abbreviation for the function name and standard variable names for the input and output. These are the most common names, but you could use others if you wanted to. Okay, so that’s the basic notation, but how did the equation get changed to f(x) instead of ‘y’? Well, it comes from the idea that if two things are equal in math, you can substitute one thing for the other. Since we’ve agreed on this general notation for a function, f(x)=y, that means you can use f(x) or ‘y’ interchangeable. Either one can represent the output set of a function. But if they’re interchangeable, why would you use the more complicated f(x) when you could just use ‘y’ instead? Well, using f(x) highlights the fact that you’re dealing with a function with a specific input variable and not just an equation. And… it gives us a handy notation for evaluating functions for specific values. For example, you could start off by saying, let the function f(x)=3x + 2. Then you could then ask someone to evaluate the function for the input value 4 by saying what is f(4). That means you’ll substitute a 4 in place of any ‘x’s that are in the function. For this function, that would mean f(4)=14. And you could do this for other values too. f(5)=17, and f(6)=20 Pretty easy, huh? Alright, so that’s what functions are in math. They’re things that relate an input value to exactly one output value. And the set of all input values is called the domain while the set of output values is usually called the range. In algebra, functions typically come in the form of equations that can be graphed on the coordinate plane by treating the input and output values as ordered pairs. Of course, there’s a LOT more to learn about functions, but this basic introduction should help you get started working with them in Algebra. Don’t forget to practice using what you’ve learned in this video by doing some exercises. As always, thanks for watching Math Antics and I’ll see ya next time. Learn more at www.mathantics.com