Mathematics Year 7: Functions: every object has exactly one image
Mathematics, Year 7 (Portugal): A function is a correspondence where each object of the starting set is linked to exactly one image, and the same function can be shown as arrows, a table, an expression or a graph.
Every object, exactly one image
A correspondence links elements of a starting set to elements of an arrival set. It is a function when every element of the starting set is linked to exactly one element of the arrival set. The element you start from is called the object and the one it is linked to is its image. Example: link each pupil in {Ana, Rui, Eva} to their birthday month: Ana → March, Rui → July, Eva → March. Every pupil has exactly one month, so this is a function. It does not matter that Ana and Eva share the same image.
When one quantity decides another
Many situations have two quantities where one decides the other: the hour decides the temperature on the thermometer, the number of apples decides the price to pay. You can only predict when each starting value gives a single result. If a rule gave two results for the same object, you would not know which to pick, and you could not calculate or draw anything. That is why mathematics studies the correspondence «one object, one image». Going the other way is a different question: 15 °C may have happened at several hours.
One function, four ways to show it
Rule: triple it and take away 2, with objects in {1, 2, 3, 4}. Expression: f(x) = 3 × x − 2, so f(1) = 1, f(2) = 4, f(3) = 7 and f(4) = 10 (read «f of 4 is 10»). Table: x = 1, 2, 3, 4 and f(x) = 1, 4, 7, 10. Arrows: 1 → 1, 2 → 4, 3 → 7, 4 → 10. Graph: the points (1, 1), (2, 4), (3, 7), (4, 10). The domain is {1, 2, 3, 4}, the set of objects. The set of images, {1, 4, 7, 10}, is the range; the arrival set can be larger, for example all the natural numbers.
A repeated image does not spoil a function
Many people think a function cannot repeat values, so they reject the table x: 1, 2, 3, 4 and y: 6, 9, 9, 12 because 9 appears twice. But it is a function: each x has exactly one y, and two different objects (2 and 3) may share an image. What spoils a correspondence is the opposite: one object with two images. In the table x: 1, 2, 2, 3 and y: 5, 6, 8, 9 the object 2 is linked to 6 and to 8, and nobody knows which is «the» image of 2. Look at the objects, not at the images: each one needs one and only one.
Thermometers, barcodes and spreadsheets
A thermometer that records the temperature every hour defines a function: each hour has exactly one temperature. The reverse is not true: 15 °C may have appeared at 9 h and at 18 h, so the temperature does not decide the hour. In a shop, the barcode of a product leads to a single price in the system. In a spreadsheet, a formula such as =3*A1-2 gives one result for each value of A1. Wherever the result is unique, there is a function behind it.
Test what you learned
A correspondence goes from a set A to a set B. What must be true for it to be a function?
- a) Every element of B must be the image of some element of A
- b) Every element of B must come from exactly one element of A
- c) Every element of A is linked to exactly one element of B
Correct answer: c) Every element of A is linked to exactly one element of B — Yes: each object has one image, no more and no fewer. The condition is about the starting set, not about the arrival set.
The function g is defined on {1, 2, 3, 4} by g(x) = 5 × x − 3. Which object has image 12?
- a) 3
- b) 57
- c) 9
Correct answer: a) 3 — 5 × 3 − 3 = 12, so 3 is the object whose image is 12, and 3 belongs to the domain.
Which table does NOT represent a function of x?
- a) x = 1, 2, 3, 4 → y = 6, 9, 9, 12
- b) x = 1, 2, 2, 3 → y = 5, 6, 8, 9
- c) x = 1, 2, 3, 4 → y = 7, 7, 7, 7
Correct answer: b) x = 1, 2, 2, 3 → y = 5, 6, 8, 9 — The object 2 appears with two images, 6 and 8. One object with two images is exactly what a function does not allow.
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