The five functions in the world of math are very important. They are one of the main reasons why math is such a smart subject. There are 5 functions that most students need to know. They are: addition, subtraction, multiplication, division, and exponents. A function is a relationship that associates each element in one set, called the domain, with exactly one element in another set, called the range.

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## Functions

In this algebra basics lesson, we’re going to learn about functions. Outside of the realm of math, The word function refers to what something does. But in math, the word function has a more specific meaning. In math, a function is 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 collection 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;

(1,2,3,4,5) (a,b,c,d,e)

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 twenty-six elements, While a set of all integers has an infinite number of elements.

So a set is just a collection of things, and a function relates one set to another. But how exactly does it do that? To understand how functions work, it will help if we start by naming the two sets, the input set and the output set.

### Input and Output Sets

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 unique math names. The input set is usually called the DOMAIN, and the output set is generally called the RANGE. And it’s common to see some or all of the functions, inputs and outputs listed in what we call a function table. A function table has typically two columns, one on the left for the input values and one on the corresponding output values.

The function itself is often written above the function table and 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. If we input a triangle into the function, the output will be three, and if we input square, the output will be four. If we input the Pentagon, the output will be five and so on.

So this function 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 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 a set of numbers that we can input the domain and Y as a set of numbers that we get as outputs the range, what we have is a straightforward 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 try inputting one, two and three as values of X and see what outputs we get for our table.

If we input the value one, In other words, if we substitute the value one for the X in our equation, then we get Y equals two times one, which simplifies to Y equals two;

Y = 2X

Y = 21

Y = 2

And since Y is our output variable, we put a two in the output column next.

y = 2x

INPUT | OUTPUT |

X | Y |

1 | 2 |

2 | |

3 |

If we input the value to add to our function, we get Y equals two times two, which means Y equals four.

Y = 2X

Y = 22

Y = 4

So the output value is four and last. If we put the value three into our function, we get Y equals two times three, which means Y equals six.

Y = 2X

Y = 23

Y = 6

So the output value is six. 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 two times X the input.

y = 2x

INPUT | OUTPUT |

X | Y |

1 | 2 |

2 | 4 |

3 | 6 |

### Limitation

So we’ve seen some examples of functions that relate inputs to outputs, but there’s a vital 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.

y^{2} = x

INPUT | OUTPUT |

X | Y |

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 saw 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 for the positive or principal root? But we also have the negative root as a solution. If X equals four, then Y equals two, and Y equals negative. Two are both possible solutions to the equation. Y squared equals X. So, in this case, each value of X that we have put into the equation will get two Y values as outputs. Can a function do that? You see, functions aren’t allowed to have what we call one too many relations where one particular input value could result in many different output values. One too many relations indeed 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 relate 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 equals two X qualifies as a function because you’ll always get just one number as an output no matter what number you put in. But the equation Y squared equals X does not qualify as a function because a single output can produce more than one output.

Let’s look at another simple algebraic equation to see if it’s a function Y equals X plus one. Again, the X values will be inputs, the domain and the Y values will be the output range. Let’s quickly generate a function table for a few possible input values like the integers, negative three through positive three.

y = x +1

INPUTX | OUTPUTY |

-3 | -2 |

-2 | -1 |

-1 | 0 |

0 | 1 |

1 | 2 |

2 | 3 |

3 | 4 |

You may notice that each row of this function table is just an ordered pair. It’s an X value followed by a Y value. We could even rewrite all the inputs and outputs to perform 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 a linear function.

In Algebra, there are many 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 can tell their functions just by looking at their graphs because they all passed the vertical line test.

Remember how functions aren’t allowed to have more than one output value for a particular input value? The vertical line test helps us see if a graph has any of that one too 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 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 that 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.

So all of these graphs pass the vertical line test and our functions. But what’s an example of a graph that doesn’t pass a 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 a vertical line does not intersect 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 the 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.

### Function Notation

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 equals two X (y=2x), and Y equals X plus one (y=x+1), but you’ll often see these same functions written like f(x)=2x or f(x) =x+1 instead. But why did the variable y get replaced with that F parentheses X thing? And what does that even mean? Well, it turns out that a common way to represent a function is this. This notation 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, f(x) = y.

The problem with this notation is that you could easily misinterpret it as a variable F being multiplied implicitly by a variable X to answer 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 a function. It would be a lot more straightforward if mathematicians used the entire work function as the name and then use the names input and output instead of X and Y. These two notations mean 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.

So that’s the basic notation, but how did the equation get changed to F of 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 of X equals Y, that means that you can use F of X or Y interchangeably. Either one can represent the output set of a function. But if they’re interchangeable, why would you use the more complicated F of X when you could use Y instead? Well, using F of X highlights the fact that you’re dealing with the 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 by saying that F of X equals three X plus two f(x)=3x+2. Then you could ask someone to evaluate the function for the input value four by saying what F of four is? That means you’ll substitute a four in place of any X’s that are in the function. For this function, that would mean F of four equals 14.

f(4) = 34 + 2 =14

And you could do this for other values to F of five equals seventeen and F of six equals twenty.