As a whole, mathematics has adopted a standardized way of communicating about functions and sets of operations. Mastering this notation early on in one’s algebra career can open doors to understanding many future topics throughout math. Misunderstanding this notation, unfortunately, can leave students floundering to understand from the get-go. Hopefully the paragraphs below provide a foundation of notation that becomes a launching pad for your algebra skills rather than a stumbling block.
This was originally published on mathteacherbarbie.com. If you are reading it elsewhere, you are reading a stolen version.
Naming a function
The image illustrates a function as a production machine. In order to produce anything, a machine needs inputs, or raw materials. The machine itself then does something to those inputs and ultimately creates the final product, or an output.
We need a way to refer to the machine itself. In math, that “machine” is really just a set of instructions. For example, the “square root function” or “square root machine” takes inputs and spits out the square roots of those inputs: $25$ becomes $5$, $2$ becomes $\sqrt{2}$ or $1.41…$.
Sometimes that set of instructions can be hard to explain easily, so we give the function machine itself a name. A common name is $f$, short for “function.” However, you might see other letters, symbols, or even longer names. For example, FIRSTNAME is a descriptive function name, implying that the output will be someone’s first name. These types of function names are often found in spreadsheets and computer programming; in math we tend toward shorter (single-letter) names, but don’t have to stick to that.
Function names in this sense are used similarly in math to both spreadsheets and computer programming.
Argument of a function
When we attempt to describe a function, especially in the form of a formula, we need a way to describe “the input.” The challenge is that the input might vary or change. So we assign a variable to represent that input generically. Then we know that every time that variable shows up in the formula, we substitute in whatever input we’re working with at the moment. That variable is called the argument of the function.
In math, the argument of a function is the collective input variable. (This is different from computer programming, which reserves the word argument for specific inputs.) We can change the representation of the argument (e.g., we can use a variable other than $x$ if we want), but this change doesn’t impact the steps the function takes or, ultimately, the output that gets assigned to any specific input.
“Inputs” vs “Outputs”
The input variable is also called the independent variable, especially when there is a real-world context around the function. This is because the ultimate result, the output, usually depends upon which input we choose but we can independently choose which input to use. Have you ever heard the acronym GIGO (garbage-in, garbage-out)? This is a specific real-world meme about this idea. What a production machine produces depends on the raw materials that are put into it each time.
The output variable is similarly called the dependent variable since its value depends upon which input is sent through the machine.
We’ve seen these inputs/outputs (independent and dependent variables) already in algebra. Most often we’ve called them $x$ and $y$, respectively, though sometimes they’ve gone by other names.
A Note About Parentheses
We use parentheses — ( and ) — a lot in math. It can be really confusing when one pair of symbols means so many different things. Unfortunately, as users of modern mathematics, we’re kind of stuck with them. As you work with math, contextual clues will help you identify which meaning to use each time you encounter them. Here are the main four you will see in early- to mid-algebra.
One of our first encounters with parentheses in mathematics is as a “grouping symbol” in the order of operations. The order of operations sets forth a standard order in which arithmetic operations (such as multiplication/division, addition/subtraction, and exponentiation) are performed. While this order is at least partially arbitrary, there is some logic behind it. Nevertheless, the role of parentheses in this process is to disrupt the order. If we actually need to perform the addition in a problem first, before we tackle the multiplication, then this is communicated by enclosing the addition inside parentheses — grouping the numbers involved in the addition together and using parentheses to indicate both that group and its order in computation.
$3\cdot 4+5$
$=12+5$
$=17$
$3\cdot (4+5)$
$=3\cdot 9$
$=27$
We’ve used a lot of symbols for multiplication throughout our math career*. In middle elementary (at least in the U.S.), we first learn to multiply using the symbol $\times$ between two numbers. Then in prealgebra, we start to replace that symbol with $\cdot$, and as we learn the order of operations, the grouping symbol parentheses sort of evolves into implying multiplication when no other operation is indicated.
Turns out, as we move through math, multiplication becomes a sort of “default,” or “primary” operation throughout our work. Partially this is because we live in a multi-dimensional world, and multiplication brings that dimensionality into mathematics (for example, 2-dimensional area = length $\times$ width and 3-dimensional volume = length $\times$ width $\times$ height). Further, the footnote at the bottom of this section even refers to a time in mathematicians’ education where there are more than one type of multiplication! Most of us won’t have to think about different types of multiplying, but I bring it up to emphasize that it’s such an essential and integral arithmetic process that it has its own story and trajectory in mathematics history.
Because of this importance in using multiplication to describe our world, when another operation is not explicitly given, we default to multiplication.
$3(4)+5$
$=12+5$
$=17$
$3(4+5)$
$=3(9)$
$=27$
*I can’t speak to why we move from the $\times$ symbol to the $\cdot$ in prealgebra. Perhaps that’s a topic for future research and another post! I can say that the $\times$ symbol reappears in later courses taken by math-major undergrads who start investigating structures that can be multiplied in different ways, as a means of differentiating those ways of multiplying.
Using parentheses to state a point on a graph is actually an application of a broader math principle of using parentheses to enclose lists.
At this point in your algebra journey, it can be useful to know three main symbol pairs that are used to enclose lists of numbers or other items in a group. A few students may have encountered the fourth symbol listed as well, though it’s not a standard part of the curriculum until multivariable calculus. Note here that the first three sets are also used in order of operations. **The context below only refers to their use in writing mathematical lists. If it’s not a list (e.g., items separated by commas), then they essentially mean the same thing as each other (e.g., grouping symbols).**
- Parentheses, ( ), around a list of numbers or items indicates that they are listed in a very specific order, and that the order communicates something about them and/or their relationship to each other.
- Square brackets, [ ], are similar to parentheses (or “rounded brackets”) in that order matters and the order of the items itself communicates something about their relationship.
- “Curly brackets” or “braces” in a mathematical list, { }, indicate a group of items or numbers that can be listed in any order and still mean the same thing. The order of the items listed inside can be changed and mixed up without impacting the meaning of the group.
- “Pointy brackets”, < >, also indicate that order matters and the order of the numbers inside indicate their relationship to each other. However, it is generally reserved for much more specific relationships such as vectors and other ideas that indicate a sense of movement and direction.
We use parentheses around ordered pairs (also called coordinate pairs or points) because the order of the numbers communicates something. In graphing, the pair $(3,2)$ indicates that the point we plot is $3$ units to the right of the origin and $2$ units “up” from the origin. The ordered pair $(2,3)$ is a completely different point: $2$ units right and $3$ units “up”.
In function terminology, the ordered pair $(x,y)$ refers to an (input,output) pair, where the first coordinate represents the input, or independent variable, and the second coordinate represents the output, or dependent variable. In our example $(3,2)$ this means that if we put $3$ into our function machine, we will get an output, or final result, of $2$ after the machine does its work.
Now we’re introducing yet another meaning of parentheses: to indicate the argument of a function. (In some contexts, especially those heavily influenced by computer programming standards, we use square brackets for this rather than parentheses. However, most algebra textbooks will stick to parentheses.)
In our formalized function notation, we can place the argument directly next to the function name, enclosing the argument in parentheses: $f(x)$. (Read this as “eff of ecks”.) In this context, where we have a function called $f$, this does not mean multiplication. It can be confusing since it looks just like multiplication, but you’re looking for clues that you’re dealing with a function $f$. Perhaps the function is defined somewhere earlier. Perhaps the word “function” or “evaluate” is used in the directions.
There are advantages to this. One is that when we “substitute in” to evaluate the function at specific inputs, we can substitute in with the parentheses. Then, after subbing in, the parentheses act just like they do in the order of operations on the side where you’re doing the computations (not the side where the function notation stands).
Another advantage is that we can change the argument, substitute in specific input values, or even chain functions together without needing new notation. Chaining them together is called composing functions. I’ve illustrated it below, but in general it’s a topic for another day, after you’re more comfortable with the basics of functions.
Putting it all together
When all of these ideas are put together in one symbolic form, $f(x)$ (read: “f of x”) acts as a unified symbol that represents the output when input x is put through the f machine. We need the whole set here — the whole “$f(x)$” to stand in for the generic output. The $x$ by itself represents the input. The $f$ by itself represents the function — the actions performed. The entire $f(x)$ as a whole represents the output we get when we perform $f$ on input $x$.
I can substitute in specific values for the argument. I can change the appearance of the argument, either changing the variable or even using an expression as the argument (such as $f(3x-2)$ or “f of 3 x minus 2”). However, if you leave off any of the three pieces, functionname(argument), you’re talking about some other part of the process, not the output.
Let’s use an example: let’s let $f$ be the “multiply by 3” machine. Then we can actually compute some of our outputs for various inputs. At the same time we can use function notation to describe those same outputs.
$f(x)=3x$
function output at $x$ = formula with $x$ as argument
$f(2)=3\cdot 2=6$
function output at $2$ = formula with $2$ as input value = 6
$f(7)=21$
function output at $7$ = 21
$f(h)=3h$
function output at $h$ = formula with $h$ as argument
$f(x+5)=3(x+5)$
function output at $x+5$ = formula with $x+5$ as argument
What’s the point of function notation?
Function notation allows us to show where a conclusion came from. The full notation, such as $f(x)$, shows both which function (or set of “instructions”) created the conclusion (or output) as well as which raw materials (input) those instructions were performed on. This gives us much more information than the conclusion (or output) alone.
In this way, function notation allows us to examine the impact of changing the input values. Eventually, these ideas expand to functions that take multiple inputs, allowing us to examine the impact of changing just one input at a time.
Beyond this, it allows us to talk about multiple functions in the same conversation. Perhaps function $f$ performs one set of instructions while function $g$ performs a different set. If we had used $y$ instead of $f(x)$ and $y$ instead of $g(x)$, we wouldn’t know which $y$ we were talking about at any given time. In this way, it also allows us to both compare different sets of instructions against each other as well as to combine the sets of instructions in a variety of ways (a topic for another post).
Further, function notation allows us to write a complicated set of instructions only once and then refer to that same set of instructions over and over again by its name instead of rewriting the whole thing out. Our examples above stayed pretty simple; “multiply by 3” or $\times 3$ are pretty quick to say and/or write. But if we had a different function, such as $f(x)=\frac{32-\sqrt{4x+17}}{2x+4} +3x$, we wouldn’t want to write it over and over again each time we change the input. Instead, we can use its name, $f$, to refer to this complicated set of operations rather than risking typographical errors and/or simply using lots of time to write it out over and over.
These give one of the most common and practical applications of this idea today. When functions are used and their variables are defined carefully, the computing power of spreadsheets allows us to tweak scenarios to answer questions like “what would happen if…?” These ideas also allow general computer programs to accept user input and still run without having to rewrite the program for every possible input.
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