# 1.7.1 What are functions?

Functions represent an unambiguous relationship between input values and the output values; each input value is associated with exactly one output value.

We've now talked about how x and y values can relate in two different settings. First, we studied them in the context of equations and then we studied them in the context of graphs. Both graphs and equations are abstract representations of some relationship between the x and y values; the x and y values in turn are abstract representations of numbers, while the numbers relate back to the physical world.

Thinking of equations as fancy "input-output" machines, we can study them further as belonging to two broad groups. First, there are input-output machines that return unambiguous responses. When we put one value into the machine we get out exactly one value. For example, at a certain age you have exactly one weight; the height of a plane, t seconds after lift-off will be just one height.

Other "input-output" relationships aren't so clear. For example, if I input a student's name into my machine hoping to get out his or her teacher's name, I could very well get out different names: one for the student's English class, one for his or her math class, and yet another teacher's name for the student's Spanish class.

There's a clarity about such relationships for which one input yields exactly one output. And, it's because of this clarity that we study "unambiguous equations", i.e. relationships where one input will yield exactly one output. In fact these types of equations will be central to this course and most of your future math classes!

We call these unambiguous relationships functions. A function is a relationship in which each input has exactly 1 output; you won't have 2 outputs, 3 outputs, and you won't have 0 outputs.

A function is a relationship in which each input has exactly one output.

Explore!

Are the following functions?

Let the input be the age of a person, in years; let the output be her weight at that point in time.

The following table represents a function

x y
1 2
1 4
2 7

The following diagram represents a function:

We typically use the symbol f(x), read as "f of x" to represent the name of the function (f in this case) and the variable that we're inputting into the function (x in this case). Note that we can plug in numbers or even other functions for the "x" in our rule. For example, if our function is f(x)=x^2+3, the symbol f(4) means tells us to plug in 4 for x, to get 42+3=19. On the other hand, if we want to evaluate f(x+4), this would become (x+4)2+3.

We also give special names to the input and output values. The Domain of the function is the set of all legal inputs; the outputs are called the Range of the function.

Explore!

For f(x)=1/x to be a function, the Domain would be all real numbers except for x= .

If you're told that the function A(w)=w(2-w) gives us the area of a rectangle based on a width w, then the Domain of this function would be all w's that are bigger than and less than .

The Range of the function g(z)=z^2+1 consists of all real numbers that are at least .

Now that we have a sense of what a function is, let's tap into another part of our brain: can we "see" if an equation is a function just by looking at it on a graph? It turns out that we can, by using a tool known as the Vertical Line Test, which we'll study in the next section.