Those googly eyes we got

We have them for good.

When we talk about f(x), we are referring to a function, not just a variable or a simple equation. The "f" represents the function itself, and "x" is the variable or input to that function. This can be confusing because we often use "f" alone to refer to the function, which can lead to misunderstandings if not explained clearly from the beginning.

In coordinate geometry, writing equations like y=mx+c feels intuitive and familiar because it directly relates to how we typically represent functions on a graph, with "y" as the dependent variable on the y-axis and "x" as the independent variable on the x-axis.

D is the distance for you, but s is it for me. It also means displacement to me. All about the context.

When we started the Coordinate geometry, y = mx + c triggers nostalgia compared to f(x) = mx+c ? When it was Pre-calculus and beyond, now that's something. Using f(x)=mx+c instead of y=mx+c may seem less nostalgic but it emphasizes the role of the function. Acknowledge the difference between an equation and a function. Here f(x) is not just any variable, but a function value dependent on x.

The notation f^2(x) can be confusing. It does not mean the function is related to logarithmic or trigonometric functions like sin or cos unless specifically stated. Typically, f^2(x) means we are applying the function f twice to x, which is a composition of the function with itself. If it's (f(x))^2, then we're squaring the result of f(x).

If the number 2 is the superscript of x, you should square x. First, note the domain of x.

If you need to square the whole function f(x) but forget to include the outer brackets, then you end up squaring only the output of f(x), not the function itself. This differs from squaring just the function f, which would imply the function applied twice to x (function composition). Leaving out brackets around the 2, when it's supposed to be an exponent of the function, is function composition, not simple squaring.

f^-1 (x) is the inverse function. This is where you reverse the mapping of f.

f alone ain’t any variable. It denotes a function and underpins the process of mapping we do. When we use "x" as the argument, it can be any element from a set, range, or a specific constant, and "f(x)" denotes the corresponding output value. This mapping involves a domain (the set of all possible "x" values), a codomain (the set where outputs reside), and a range (the actual outputs that occur).

"f" and "f(x)" aren't really the same. The function "f" refers to the mapping rule itself, while "f(x)" refers to the output for a given input "x". Mentioning what something exactly means is usually done before. That doesn’t seem important in the start but going truly forward, it helps! For example :

If we define a function as f:x ↦ ln(x), "f" is the function taking "x" and returning its natural logarithm.

To me, P(t) would be the perimeter of some figure at t.

Inside the notation "f(x)", variables like "k" or "c" are specific inputs. The output "f(x)" is calculated at each point on the graph of the function "f", where it usually appears on the left-hand side of an equation.

In everyday use, a relation binds/connects people or entities. Function is more about "What will this be doing?", the role that something is meant to play, i.e. the utility. Domain is a certain realm or a field of interest/expertise. Range is the extent of something, whether it be temperatures, vocals/music, or shooting distances, defining the limits of possibilities.

In mathematical/algebraic context, a relation is a set of values (x,y) described by the equation; a curve is an example. By definition, a function has a set of inputs (domain) and a set of outputs (range). It's a curve in x - y plane that is the graph of a function of x if and only if no vertical line intersects curve more than once. It's a certain type of relation. The domain and range of a function are the sets of x and y values that it can take as inputs and outputs, respectively.

As for comparison, they just build upon each other. Building on intuitive understandings to understand how different entities relate and affect each other. You can quantify them & examine if you can generalize the observations and optimize the experience, all while sprinkling a bit of math in there. It is all analytical. You’re trying to make the abstractness more tangible and relatable, which is easily the most recognized way of establishing and describing relationships and constraints within and between various entities. Isn’t the whole world full of relations (music, economics, societies, machines) that we want to comprehend and make use of ?

All algebraic equations can be called relations because they establish relationships between variables like x, y and z, etc. A relation that assigns more than one output to a single input is not a function. For instance, x^2 + y^2 = 16 is a relation. This is not a function, because it fails vertical line test that judges if a relation is a function or not.

Mathematics is a language that allows for a high degree of personalization and interpretation. How you use and interpret mathematical notations and functions can vary widely, adding richness to it. This flexibility makes me think of how Math epitomizes self-expression in problem-solving.