Q.Program 10-3: Write a program to compute the area and circumference of a circle using a function.

What is a Function? — From Intuition to Definition

Think of a function as a machine that takes an input and gives you exactly one output. You put something in, the machine does something to it, and out comes a result.

A simple example: a squaring machine. You feed it the number 3, it gives you 9. You feed it −2, it gives you 4. Every time you put in the same number, you get the same output. That's the core idea — consistency.

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Real-life intuition

Imagine a vending machine. You press button A1 — you get chips. You press B3 — you get a cold drink. Each button gives exactly one item. If pressing A1 sometimes gave chips and sometimes gave a chocolate bar, the machine would be broken. A function works the same way: one input, one output.

Now contrast this with a non-example: a machine that takes a person's name and returns their birthday. That's fine — each person has one birthday. But if the machine took a birthday and returned a person's name, it would break: many people share the same birthday, so one input would give multiple outputs. That's not a function.

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The precise definition

Let's unpack that sentence piece by piece.

"From a set $A$ to a set $B$" — $A$ is called the domain (all possible inputs), and $B$ is the codomain (the set that contains all possible outputs). The function doesn't have to use every element of $B$; it just has to stay inside $B$.

"Each element $x$ in $A$" — every single input must have an output. No input is left out. If the domain is all real numbers, then every real number must produce something.

"Exactly one element $y$ in $B$" — this is the heart of the definition. For a given $x$, there is one and only one $y$. No ambiguity, no multiple choices.

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How to check if something is a function

Given a rule, ask two questions:

1. Does every input in the domain produce an output? (No gaps.)
2. Does any input produce more than one output? (No ambiguity.)

For example, $y = \sqrt{x}$ with domain $x \geq 0$ is a function — every non-negative $x$ gives exactly one non-negative square root. But if you allowed both the positive and negative square roots, it would not be a function, because one input would give two outputs.

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Why this matters …