A function is not a formula | one thing in, one value out — and the vertical line test

You press the same button on a vending machine and some days a cola comes out, some days an orange soda. Would you put your money in? One thing in, one thing out — at school they called that rule a function. But the sentence a textbook uses to define a function has no formula in it. There is exactly one condition: for each thing you put in, exactly one thing comes out. Which means last episode's sine was a function too — with no formula at all, just angles and values written down in pairs. 📍 Chapters 0:00 Same button, different drink 0:20 It was never the formula 0:50 Only one condition 1:22 What a function is not 2:02 The sine from last time 2:33 One function, four outfits 3:19 The name called a function value 4:05 Turn squaring around 📌 In this episode · The definition of a function contains no formula — "one thing in, one thing out" is all of it · Nothing coming out will not do, and two will not do. The other direction is free · Domain = what you are allowed to put in · Range = the values that *actually* come out · The vertical line test — if no vertical line meets the picture in two points or more, it is a function (meeting it zero times is still a function) · The whole circle is not the graph of height as a function of the horizontal position. The upper half is · Last episode's trig table: being able to write exactly one value beside each angle is the evidence · One function written four ways — words, table, graph, formula. The formula is just the shortest · Why we get to write f(3) at all: three already has exactly one value waiting for it · x² = 4 having two answers is not a function handing out two values — we searched backwards · Run squaring backwards and one splits into two → the square root sign is us choosing the branch that is zero or above. That is also why the square root of zero is zero 🔗 Earlier EP.01 One map from the counting numbers to the complex ones EP.02 Imaginary numbers are not imagination, they are direction EP.03 Walk π along the circle and you are halfway round EP.04 A 100 percent grade is 45 degrees EP.05 Similarity — erase the size, keep the shape — Mathiter is a math practice app for international-school and study-abroad students. Practice that meets your level → https://mathiter.com #math #function #domain #range #verticallinetest #mathconcepts #middleschoolmath