mathematics / functions
Functions
One input, one output, every time. What a function is, how a graph records one, and the two families that grow and shrink unlike anything else.
Three pages, in order
A function is the object everything else in mathematics is written in terms of, and it is usually introduced in half a sentence before the exercises start. That is backwards. The definition has one clause that matters, one input gives one output and always the same one, and nearly every later difficulty is that clause being enforced. These three pages build the idea from a machine you feed numbers into, show that a graph is a record of every pair rather than a picture of a formula, and then take the two families whose behaviour nothing else matches. Every figure computes what it draws from the controls you set, so no picture on these pages was stored.
The Function
A function is not a formula. It is a rule that gives every input it accepts exactly one output, and the three things worth knowing are which inputs it refuses, which outputs it never reaches, and what a graph actually records.
Drawing the Graph
A curve on a screen is a short table of values with the gaps between them guessed. Six shapes cover most of what you will meet, and three numbers turn any one of them into a whole family, so a family is one curve moved rather than a list to learn.
Exponentials and Logarithms
Exponential does not mean fast. It means the rate of growth is set by how much there already is, which is why a doubling pile loses to a pile adding 3 for the first three steps and then leaves it behind for good. A logarithm is the exponent you were missing, and turning multiplication into addition is what it was built to do.