Functions

The world is very "functional".
Exs. At any one x time you / company / economy is worth a y dollars. and you are at one y location, moving at one y speed, weighing a y pounds, etc.

Equation of two variables, say x and y, that is or can be put into the form y = expression_of_x
The expression_of_x is the definition of the function.
A function has a name, most often the generic ƒ or g or h or a word like height, profit, cost etc.
A function is a function of some variable, most often the generic x or t or whatever.
   So ƒ(x) means the function named ƒ of the variable x. x is the argument or input of this function.

Some functions:
Ex.    ƒ(x) = 7x²+3x+4
Ex.    ƒ(x) = 3x+4
Ex.    ƒ(x) = ∛(x²+3x+2)
Ex.    h(t) = -4.9t²+20t
The "=" here doesn't really mean an equation but is used to denote the definition of the function. Perhaps a : or a := would be better?
y is essentially the same as ƒ(x) so y = ƒ(x)

Using ƒ(x) = 3x+4, evaluating the function for a particular value for x: ƒ(0) = 4, ƒ(1) = 7, ƒ(-3) = -5, etc.
The value or output of the function is the result of the evaluation.

Function as a machine: input to it is the x value, it evaluates the expression using the inputted number, then outputs the result, i.e. the y.

          f
       ▁▁▁▁
       ▏     ▕
x ---> ▏     ▕---> y
       ▏     ▕
       ▔▔▔▔  

The graph of a function shows the points (x,f(x)) for each x value in the domain of the function its corresponding y value. Same as the solution set of the equivalent y= equation.
A function's graph always goes to the right "forwards", it never "reverses" or backtracks to the left, never loops, and is never vertical.

A function's graph passes the vertical line test: no vertical line crosses the graph more than once because a function assigns/pairs/maps/associates each x value with only one y value, i.e. each x number results in just one y value.

A function's domain might be only a subset of R. i.e. not every x value is in the domain.
A function's range can be from a single number to all of R. i.e. not every y value is in the range.
A function has one y-intercept if 0 is in its domain. Else no y-intercept.
A function can have any number (0 to ∞) of x-intercepts.
A function might have a symmetry: across the y-axis (even function) or about the origin (odd function).
   f(-x) flips over Y-axis.
   -f(x) flips over X-axis.
   -f(-x) flips over both X-axis and Y-axis; equivalent to 180° rotation about origin.
   If f(x)=f(-x), then f is Even (symmetric across the Y-axis). Also: -f(x)=-f(-x)
   If f(-x)=-f(x), then f is Odd (symmetric about the Origin). Also: f(x)=-f(-x)
A function can be positive on an interval, negative on an interval, or 0 on an interval.
A function can be increasing on an interval, decreasing on an interval, or flat on an interval.
A function might have turning points.
A function might have local or global/absolute extrema (mins or maxes).
A function might have: vertical, horizontal, oblique asymptotes.
A function might have discontinuities.
A function could be periodic.
A function might have concavity on an interval and might have changes of concavity (inflection points).

Classes/groups/categories of functions:

Algebraic functions: polynomials +-*/^√
Transcendental functions: beyond algebraic functions, e.g. exponential, log

Functions of two or more variables.
2 variables, say x and y: f(x,y) = expression_in_x_and_y
Result/output could be z, so f(x,y) = z
A function of two variables' solution set is triples (x,y,z) that graph in 3D, etc.
Ex.    BMI is a function of weight and height, in metric: bmi(w,h) = w/h²
Ex.    Area of a triangle is a function of base b and height h: A(b,h) = ½bh
Ex.    Future value is a function of a principle P of money, with interest rate r, compoundings per year n, and time t in years: FV(P,r,n,t) = P(1+r/n)^nt    A function of 4 variables.