Concept And Notation

Math is essentially the study of patterns. Or so I'd like to believe. In math, and science for that matter, we are searching for patterns and attempting to describe them. In math we tend to do this, but it is actually a somewhat new idea, with equations or graphs. Historically the pattern was to describe patterns and mathematical concepts in words. The formalization of math in equations while often seen as all there is to math, its far from true, is actually a gift. We can now write something in symbols and virtually any other math student in any other country will (or should be) able to know what we are saying. The ambiguity is gone…

With that said, I am a big fan of students describing mathematical ideas in words, equations may dominate the typical math classroom and most books but they are far from the be-all and end-all of math.

One of the key conceptual developments in math is the idea of the function. Not the concept of the equation but the concept of the function. It truly is an abstract concept, just try explaining it to someone who doesn't already understand it, but once you understand it then its pretty easy and even useful… Especially in computer programming.

So enough rambling lets get to it.

### The concept of the function

Functions in the mathematical sense are no more than a set of instructions on how to operate a mathematical machine. The machine gets an input (something goes in) the function or instructions tell the machine what to do with the input(s) and the result is the output. i.e. the cow goes in and the steak comes out. The machine does the rest…

In math the input is most often a number or a variable. The function operates on the input. For example a function could add 2 to any input. The input could be any real number. For example the input could be 4 and the output would be 6. If the input was -12 then the output would be 10. In function notation the above function would be written as:

(1)
$$f(input) = input +2$$

In function notation the part in the parentheses on the left is the input, often called the "argument" in math lingo. No matter what is there the function does the same thing to the input. In math classes function notation is often written as:

(2)
$$f(x) = x+2$$

This is read as "f of x equals x plus two." This can be a bit confusing to students as "x" is often used as a coordinate on a graph or holds other mysterious properties. The "x" in function notation is not special it is just a place holder. The function above can be written as:

(3)
\begin{align} f(\square)=\square +2 \end{align}

And it means EXACTLY the same thing as the previous two expressions! The only difference would be how the expression is read. In this case the expression is read as "f of square equals square plus 2." The symbol used for the argument does not matter one bit.

Lets look at some of the confusing bits of function notation, or at least where students can get jammed up. Lets define a new function, this function will double the input and then subtract 3. This translates into math symbols as:

(4)
\begin{align} f(\square) = 2 \cdot \square -3 \end{align}

If our input is 4 then the function is written as $f(4)=2 \cdot 4 -3=5$. If the input is x then the function is written as $f(x)=2\cdot x -3$ in this case there is no numerical value to the function… Now we get a bit trickier. What if the input is not x but rather "x+2"? The function is then written out as:

(5)
$$f(x+2)=2(x+2)-3$$

This can be simplified to be written as $f(x+2)=2x+4-3=2x-1$. Notice that the function doubles the entire input not just the "x" or the "2." Where function notation can be particularly confusing to students is when the function is defined in terms of x. Such as:

(6)
$$g(x)=3x^2 + 1$$

Note that the "f" in function notation is not special. The part of the function notation before the parentheses is the name of the function. The name does not need to be only one letter, it can be anything and should often be chosen to be helpful to the person using the function.

The function squares the input then multiplies by 3 and finally adds 1. After the function is defined; a teacher, textbook or test may then refer to the function of $g(2x)$. What this means is that the input of the function is 2x. So if we follow the instructions of the function…

(7)
$$g(2x)=3(2x)^2 +1$$

Its important to notice and understand that the entire "2x" is squared not just the 2 or the x!

### Why Bother?

So why is function notation important? As mentioned above functions are essentially a compact way of writing specific mathematical instructions. Function notation is very general, that is a function gives directions for any input… this is particularly important for computer programming when a programmer may want to reuse the same set of instructions over and over but for a variety of different inputs. Function notation also makes transformations easy (or at least easier) to see and understand. If you are feeling confused and frustrated go find your teacher (or post questions below) and keep fighting to understand. Function notation is not going to go away. It is not something that you can ignore and hope to be successful.

### Functions vs. Relations

In math a function is defined very carefully, not just any equation is a function. If two sets of numbers are related to each other (think the x and y values for a graph) they are most often categorized as either a relation or a function. For example the age of a person and their height. Or the distance a person is from their bed and the time of day…

Both a relation and a function can be described with equations or graphs. All functions are relations, but relations are not necessarily functions. Lets try to flesh that idea out a bit.

Definition of a Function: In order to be a function there can only be one output for every input. At a particular time of day (the input) a person is a given distance from their bed (output). A person can not be both 1km and 3 km away from their bed at the same time. Since for each time there is only one distance then this is an example of a function. This relationship between the input and output is often referred to as "one-to-one," that is for each one input there is only one output.

This definition can be a bit confusing and determining whether something is a function can be challenging. However, if the numbers or equation is graphed then determining whether something is a function is made much easier by using what is often referred to as the "vertical line test." If a vertical line can be drawn such that it crosses the graph more than once then it is not a function.

The graph above is an example of a relation. It is possible to draw a vertical line such that it crosses the function more than once.

The graph above is an example of a function. At no point on the graph is it possible to draw a vertical line that crosses the graph more than once.