Quarry School

Math, taught from zero.

Free courses that assume nothing. Every lesson says the idea in plain words, shows a picture, gives the reason before the rule, works an example one step at a time, and names the mistake most people make.

See the two courses Read a sample lesson

A sample lesson

What a function is

Plain words

A function is a rule that takes an input and hands back exactly one output. Think of a vending machine: press the same button and you get the same snack every time.

Picture

4 input multiply by 3, then subtract 2 the rule f 10 output
Put 4 in. The rule gives 3 × 4 = 12, then 12 − 2 = 10. One input, one output.

The rule

f(x) is the output of the rule f for the input x.

Read the whole lesson, with its worked example and its trap

Two courses, free

Lessons go live one unit at a time. Each course page lists its units and says which are written.

Trigonometry

Angles and how to measure them, the six functions, their graphs and their identities, from the first picture of an angle to sinusoids.

Six units. Free.

College Algebra

Functions from the ground up: notation, domain and range, how graphs behave, transformations, and lines.

Eight units. Free.

How every lesson runs

The same seven parts, in the same order, in every lesson of every course.

  1. Plain words. The idea in everyday language, before any symbol appears.
  2. A picture. A diagram for every idea and for every new word.
  3. Why. The reason the rule is true, so there is nothing to take on faith.
  4. The rule. Stated once, exactly.
  5. How. The steps, in the order you do them.
  6. A worked example. Every step shown with its reason, then the answer checked.
  7. The trap. The common mistake, and how to catch it in your own work.

A sample lesson: what a function is

This is a full lesson at its real length. It needs nothing but arithmetic.

Plain words

A function is a rule that takes an input and hands back exactly one output. Think of a vending machine: press the same button and you get the same snack every time. Two buttons may give the same snack. One button never gives two different snacks.

Picture

4 input multiply by 3, then subtract 2 the rule f 10 output
Put 4 in. The rule gives 3 × 4 = 12, then 12 − 2 = 10. One input, one output.

Why

Because each input has exactly one output, you can name the output before you compute it. That is the whole point of the notation: f(4) names one number, the one the rule produces from 4.

The rule

f(x) is the output of the rule f for the input x.

How

To find f of a number, put that number in parentheses everywhere x appears, then do the arithmetic.

Worked example

f(x) = 3x − 2. Find f(4).

  1. Replace x with 4, in parentheses: f(4) = 3(4) − 2. The input goes everywhere x was.
  2. Multiply first: 3(4) = 12, so f(4) = 12 − 2. Multiplication comes before subtraction.
  3. Subtract: 12 − 2 = 10. One input has produced one output.

f(4) = 10

Check: the machine in the picture turns 4 into 10.

The trap

f(4) does not mean f times 4. The parentheses hold the input. Read it aloud as “f of 4”.