Tropic of Calculus

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Here’s a chapter section from the “math in the vernacular” textbook that I’m not writing, “Tropic of Calculus”.

(Ed note: The previous section would have covered the definition of functions and the use of the Vertical Line Test)

5.1 Function Notation

If you were raised in the United States, you probably have a first name, middle name, and last name. But unless you’re in trouble and your mother is lecturing you, people probably don’t call you by your full name very often. Instead of being called “James Tiberius Kirk”, your friends probably call you “James”, “Jim”, or even “Captain”. It’s just more convenient to refer to people with a nickname or a shortened version of their name, rather than using their full name every time you want to talk about them.

The same is true of functions. (Remember those? In the previous section we talked about the specific type of mathematical equation we call “functions”.)
You might have a function with a rather lengthy “full name”. For instance:

y = 404 x^7 + 1690 x^5 + 43 x^4 + 15 x^3 + 2020 x^2 + 5133 x + 19

If you had to repeat that entire thing every single time you wanted to talk about that function, you’d be exhausted before your class was even half over. So mathematicians have developed a “nickname” for specific functions, a shorthand for referring to functions.

Using the function from above, the “nickname” looks like this:

f(x) = 404 x^7 + 1690 x^5 + 43 x^4 + 15 x^3 + 2020 x^2 + 5133 x + 19

f(x)? What? Okay, let’s break it down.

This does not mean “f times x”. I know it looks like it does. It’s confusing that way. On behalf of mathematicians everywhere, I apologize.
Nor is ” f ” in ” f(x) ” shorthand for the word “function”. Mathematicians are just as likely to use ” g(x) “, or ” h(x) “, ” s(t) “, ” v(t) “,… the list goes on and on.

So what does it mean? It’s just a nickname for the function. If it helps, think of the ” f ” in ” f(x) ” as standing for “Frank”, or the ” g ” in ” g(x) ” standing for “George”.

What we’re trying to say is “Here’s a function. We’re gonna call it ” f “.”
And the ” (x) ” provides just a little extra information. It tells you that the independent variable for our function is an ” x “.

Here’s the “Math-to-English™” translation of a couple of examples:

Math: f(x) = x^2 + 4 x + 4
English: See that equation over there? It’s a function. And we’re going to call the function ” f “. The independent variable in the function is ” x “. And the full equation for the function is ” x^2 + 4 x + 4 ”

Math: g(x) = 1/x
English: Here’s another functions. This one is called ” g “. Yo, g. The independent variable in this function is ” x “. And the full expression for the function is ” 1/x “.

Math: s(t) = -16 t^2
English: I’m calling this function ” s “. The independent variable in this function is ” t “. And the full expression for the function is ” -16 t^2 “.

And if you’re reading this aloud, ” f(x) “, ” g(x) “, ” s(t) ” is read as “f of x”, “g of x” and “s of t”.

5.1a Independent and Dependent Variables

As we said above, for some function…
f(x) = x^2 + 4 x + 4
… ” x ” is the independent variable in the function.

Just to recap, the “independent variable” is the same thing as the “input variable”. It stands for some number you might be plugging into the equation. And if you’re graphing the function, this will be the horizontal axis for your graph.

So if ” x ” is the input, the entire ” f(x) ” is the output. Think of it like this: you plug some x into the function, do some arithmetic, and the ending result is your output. The “output variable” is the same as the “dependent variable”. And if you’re graphing the function, this will be the vertical axis for your graph.

We can use function notation to talk about the plugging in specific values into a function.

s(t) = -16 t^2

Let’s say we want to plug a ” 1 ” into this function. We’d just replace the ” t ” in the function with a ” 1 “, right? We do the same thing in the function notation.

s(1) = -16 * 1^2 = -16
1 was the input, – 16 is the output.

Similarly,

s(2) = -16 * 2^2 = - 64
2 was the input, – 64 is the output.

s(3) = -16 * 3^2 = - 144
3 was the input, – 144 is the output.

If we were graphing this function, we know the function would pass through the following coordinates:
(1, -16), (2, -64), (3, -144)

We can even use function notation to input something more than plain numbers into a function. For instance:

f(x) = x^2 + 5 x + 6

Let’s say we wanted to plug some constant into f(x). In this example, our input is ” π “, which is the number 3.14159… (it goes on and on and on…)

f(π) = π^2 + 5 π + 6 ≅31.5776

We could also plug in a “placeholder constant” that doesn’t have a specific value just yet, like ” n “.

f(n) = n^2 + 5 n + 6

We can even plug in slightly more complicated values, like ” 2m ”

f(2m) = (2m)^2 + 5(2m) + 6
= 4 m^2 + 10 m + 6

Or plugging in ” x + h ”
f(x + h) = (x + h)^2 + 5(x + h) + 6
= x^2 + 2xh + h^2 + 5x + 5h + 6

Here are some examples for you to try on your own. You’ll want to get comfortable with this topic, because we’re going to be building on it a lot in later sections.

(Ed note: domain and range, composite functions, inverse functions and transformations of functions)

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