Speak Maths · Last school years
Functions: machines with an input and an output
A function is a machine: put a number in, get a number out. Six signs describe it, how machines join, and how to run one backwards.
Draft: written by an AI helper and not yet checked by a teacher.
The alphabet
Read each sign aloud, the way it is written under it.
- Say: f of xMeans: what the machine f gives out when x goes in; not f times x
- Say: f from A to BMeans: f takes each allowed input in A to an output in B
- Say: x maps to two xMeans: the rule: input x becomes output 2x
- Say: g after fMeans: first f, then g, on the result
- Say: g after f, of x, equals g of f of xMeans: put x in f, then put what comes out into g
- Say: f inverseMeans: the machine run backwards; the minus one is not a power to divide by
The story: A machine that toasts bread
Tap a line to see its words.
- The toaster is a function f: A holds the minutes you may set, B holds the darkness levels it can give.
- Its rule: x minutes maps to darkness level 2x.
- We name the rule: f of x equals 2x.
- Three minutes gives level 6. Five minutes gives level 10.
- A second machine g adds one spoon of jam to each level: g of y is y plus 1.
- Toast, then jam, for 3 minutes: g after f at 3 is g of 6, which is 7.
- Run the toaster backwards: toast of level 10 needed f inverse of 10, which is 5 minutes.
- Likewise f inverse of 6 is 3 minutes.
- Careful: one over f of 6 is one twelfth, not 3. The minus one is not a reciprocal.
In words this story is 599 letters long; in symbols, 84.
Read it
What does it say?
Write it
Which notation says this?
The machine f sends every x to three times x.
Put 2 in f first, then put the result in g.
The inverse function of f, at 4.
The machine f halves every input.
g takes natural numbers to natural numbers.
The output of f when 7 goes in.
Each input x becomes x minus one.
Do f first, then h: h after f.
The input that f sends to 20.
Put 5 into g, then put the result into f.
The machine f triples its input: f of x is three x.
g after f, at 2, equals g of f of 2.
Build it
Tap the tiles in order. One tile does not belong.
f of 3 equals 6.
g after f, applied to x.
x maps to x plus one.
f from A to B.
f inverse of ten equals five.
g of f of two.
Challenges
Creative puzzles that use every sign of this chapter. Take your time; there is no clock.
1 The toaster table
A toaster f turns minutes (rows) into darkness levels (columns). Judge each sentence: true or false?
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A tick means f turns the row's input into the column's output.
Every input has exactly one tick, which is what makes f a function. Two inputs may share an output (2 and 3 both give b), but one input never gives two.
2 Same letters, different machines
Pair each line with what it says.
Tap a line, then its meaning.
f⁻¹(x) is not 1/f(x), and g(f(x)) is not f(g(x)): the order of machines matters.
3 Which machine first?
With f(x) = 2x and g(y) = y + 1, toast first, then jam, for 3 minutes. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
g after f means f runs first: g(f(3)) = g(6) = 7. The other order gives f(g(3)) = 8.
4 Run it backwards
What does the inverse machine give for 7?
f⁻¹(7) is the input that f turns into 7: 3 · 3 − 2 = 7. And 19 is f(7), going the wrong way.
5 Two machines in a row
Put 3 into f, then put the result into g. What comes out?
f(3) = 4, then g(4) = 8. Choosing 7 would mean running g first.
6 The square-root machine
This machine only accepts 1, 4 and 9. What comes out for 9?
x ↦ √x sends 9 to 3. The reading 81 squares instead, and 1/3 mistakes the machine for a reciprocal.
7 Design a machine
A machine doubles every input and then adds one. Write its rule in symbols, naming the machine f.
One way:
Either f(x) = 2x + 1 or x ↦ 2x + 1: the first names the machine, the second shows the journey of x.
8 Socks and shoes
Step s puts on socks, step h puts on shoes. “Socks, then shoes” is h ∘ s. Write the rule for undoing both: the inverse of “socks then shoes” is “undo shoes, then undo socks”.
One way:
To undo a chain, undo the last step first: (h ∘ s)⁻¹ = s⁻¹ ∘ h⁻¹.
9 The add-four machine
A machine f turns each input (rows) into an output (columns). Judge each sentence: true or false?
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A tick means f turns the row’s input into the column’s output.
Every input gains 4. The inverse goes backwards: f⁻¹(6) is the input that gave 6, namely 2, not one sixth.
10 The doubling machine
Machine f doubles. Judge each sentence: true or false?
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A tick means f turns the row’s input into the column’s output.
f ∘ f runs the machine twice: 1 becomes 2, then 4. And f⁻¹(2) asks what gave 2: that is 1, not 4. Finally 2 + 4 is 6, not f(4) = 8.
11 Chain or sum?
Pair each line with what it does to x.
Tap a line, then its meaning.
The little circle chains machines, right one first; the dot and the plus combine two separate outputs.
12 Two arrows and a colon
Pair each line with what it says about the doubling machine.
Tap a line, then its meaning.
→ goes between sets, ↦ between one input and its output; f(x) names the output and f⁻¹ runs the machine backwards.
13 Undo the halving
Machine f halves its input, so its inverse must double. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
To undo dividing by 2, multiply by 2: f⁻¹(x) = 2x. The minus one is not a reciprocal.
14 The wrong arrow
f takes natural numbers to natural numbers, and each x goes to x plus 3. One tile is wrong.
Tap the tile that is wrong.
The plain arrow → goes between the two sets; a single input travels with ↦, the arrow with a bar.
15 Which input?
Which input does f turn into 4?
x/2 + 1 = 4 means x/2 = 3, so x = 6. Check: 6/2 + 1 = 4. The value 3 is f(4), the wrong direction.
16 Take one, then square
Which runs first here, f or g? Then work it out.
In f ∘ g the g runs first: g(4) = 3, then f(3) = 9. Doing f first gives 16 − 1 = 15.
17 The fence machine
A machine p takes the side s of a square garden and gives the length of fence around it: four sides. Write its rule.
One way:
p(s) = 4s names the machine and its output; s ↦ 4s shows the journey of one input.
18 Peel, then slice
Step p peels an apple, step s slices it. Write “peel first, then slice” as one combined step.
One way:
The step done first is written on the right: s ∘ p means p, then s.
19 Seven forward, seven back
Machine f adds 7 to every input. Write f and its inverse.
One way:
Adding 7 is undone by taking 7 away, so f⁻¹(x) = x − 7 and not 1/(x + 7).
Chapter finished. The symbols are yours.
Tell your own
Tell a three-line story about a machine of your own (a rule, one input, one output), in signs only. Then run it backwards.
Kept on this device only.