# The Goal

The primary goal of this section is to clarify what is meant by the notation $y = f(x)$ and how to interpret the meaning of something like $f(x + h)$, $f(x^2)$, $f\left(\sin^2(x)\right)$ or $f(a + i\Delta x)$. In my experience, most students actually find the concepts in calculus very intuitive – it’s the notation that they find confusing. This is great news: you get the hard part (the ideas), you just need to understand how we express these ideas mathematically.

First things first: $y = f(x)$ should be read “is a function of x” notequals of x.” It’s not so much that the latter is wrong, it’s that you risk losing the most valuable content of the statement, viz. y is the dependent variable whose value is determined by x, the independent variable. It tells you: “if you know the value of x, then f is a rule that allows you to determine y.

The above should clarify what might otherwise appear to be a purely redundant statement like y = y(x). This is read as “y is a function of x, which is a statement with actual (important!) content. I think of this as equivalent to common abbreviations used in academic writing like e.g., i.e., etc. (pun intended :D): just like our notation for functions, these are shorthand expressions used to save us time when writing, but it is much more helpful to read “i.e.” as “that is” (i.e. is short for id est, Latin for that is) than just saying the letters aloud.

# Understanding the notation

There is more information in the statement $y = f(x)$: in addition to identifying the dependent y and independent x variables, it gives you the name of the function: f. Once it has been identified, if we want to talk about the function we don’t have to write f(x) each time. We can just say: “oh yeah, I was playing around with f yesterday and didn’t understand anything.”

This also means that changing the name of the independent variable does change the function itself. This is best illustrated by example:

1. If $y = f(x) = x^2$, then:
• $f(t) = t^2$;
• $f(\sin(x)) = (\sin(x))^2 = \sin^2(x)$;
• $f(a+b) = (a+b)^2 = a^2 + 2ab + b^2$;
• $f(x^2) = (x^2)^2 = x^4$;
• $f(a +i\Delta x) = (a + i\Delta x)^2 = a^2 + 2i\Delta x + i^2 \Delta x^2$;
• $f(x + h) = (x+h)^2 = x^2 + 2xh + h^2$ etc.

This suggests that we think of this function on a more conceptual level as “the function that squares its argument.” Whatever you plug in comes out squared.