Functions: a number machine · Lập trình với Martin

You put x in, y comes out

Today one of the biggest year 9 topics starts: functions. It sounds learned, but it is an ordinary number machine. You put a number x in, the machine processes it by its rule and a number y comes out.

A ticket machine is a function: you put in the number of zones, a price comes out. A taxi meter is a function: you put in kilometres, an amount comes out. A charger is a function: you put in minutes of charging, battery percents come out.

The machine’s rule is written as an equation. For example y = 3x + 5. You read: “take x, multiply by three, add five — and that is y.”

An important property: for one x, always only one y comes out. The machine does not roll a dice. Same input, same output, every time.

Today you will learn to read a function, calculate its values and build a table. Graphs come in the next lesson.

Jonáš is ordering a taxi in an app. “Pick-up 40 Kč and then 30 Kč per kilometre,” Sofie reads over his shoulder. “That is a function: y = 30x + 40. You put in kilometres, a price comes out.” Jonáš tries: “So 5 kilometres… 30 times 5 is 150, plus 40 is 190 Kč.” Sofie nods: “And the app counts it exactly the same. No magic — it just substitutes into the formula. Who can do functions can see into how prices are made.”

💡

Always translate the word “function” to “number machine” and the equation to a recipe: what happens to x before it becomes y.

Writing, substituting and a table of values

How you write a function: y = 2x + 1. The letter x is the input (what you put in), y is the output (what comes out). Sometimes you write f(x) = 2x + 1 — it reads “f of x” and it means the same.

Calculating a value = substituting. I want to know what the machine does with three. I write three instead of x: y = 2 · 3 + 1 = 7. Done: for x = 3, y = 7.

A table of values: several inputs and outputs neatly next to each other. For y = 2x + 1:

x: 0, 1, 2, 3 y: 1, 3, 5, 7

Notice the pattern: when x grows by 1, y grows by 2. That is what the two next to x does — in the next lesson it will become the slope of the graph.

What a function is not: a rule that for one input gives more outputs. “You put in 3 and 7 comes out, or maybe 10” — that is not a function, that is a broken machine.

Method for today: take the rule, substitute four different x (even zero and a negative number is fine), write a table. Who can do a table has a graph for miễn phí in a week.

Example 1: values of the function y = 3x + 5

Question: Calculate y for x = 0, x = 2 and x = 10.

Method: I substitute instead of x:

For x = 0: y = 3 · 0 + 5 = 5. For x = 2: y = 3 · 2 + 5 = 11. For x = 10: y = 3 · 10 + 5 = 35.

Table: x: 0, 2, 10 → y: 5, 11, 35.

Notice: zero is the fastest substitution in the world — everything with x disappears and only +5 is left. Always substitute zero first, it is a miễn phí check.

Example 2: a taxi as a function

Question: A taxi charges 40 Kč pick-up and 30 Kč per kilometre. Write the rule and calculate the price for 3 km and for 8 km.

Method: Input x = number of kilometres, output y = price in Kč.

Rule: y = 30x + 40.

For x = 3: y = 30 · 3 + 40 = 90 + 40 = 130 Kč. For x = 8: y = 30 · 8 + 40 = 240 + 40 = 280 Kč.

Sense check: 5 km extra should cost 5 · 30 = 150 Kč extra. And indeed: 280 − 130 = 150. It matches.

Example 3: find x when you know y

Question: The function y = 2x + 1. For which x is y = 15?

Method: Now I go the other way — I know the output, I look for the input. I substitute y and solve the equation:

15 = 2x + 1.

I subtract 1: 14 = 2x. I divide by two: x = 7.

Check: y = 2 · 7 + 1 = 15. It matches.

Answer: x = 7. This turnaround (from y back to x) is a favourite entrance-exam question — and it is just an ordinary equation.

⚠️

Do not mix up input and output. The question “what is the value of the function for x = 4” means substitute 4 for x. The question “for which x is the value 4” means set y = 4 and solve the equation. Read the question twice — they sound similar and each wants the opposite method.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

A number machine for real: substitute x into the rule and write what comes out. Write a whole number.

Practise the questions

On paper

Tables of values and one number machine of your own.

The function y = 2x + 3. Build a table for x = 0, 1, 2, 3, 4. By how much does y grow when x grows by 1?
The function y = 5x − 2. Calculate y for x = 0, x = 2, x = 6.
Invent a function from real life (charging a phone, the price of photocopying…), write the rule and calculate two values.
The function y = 4x + 1: for which x is y = 21? Solve it as an equation and do the check.

Now you 💪

You can read y = 3x + 5 as a recipe: multiply by three, add five.
You have built a table of values and you see regular growth in it.
You can also do the reverse: from a value y find x with an equation.

✅

Done when: You have two tables of values, one function of your own from life and one solved reverse question (from y to x).

What to take from this lesson

A function is a number machine: you put x in, y comes out by the rule.
For one input there is always only one output.
You calculate a function value by substituting x into the rule.
A table of values is the fastest way to get to know a function.
From y back to x you get with an equation.