Q.What is the purpose of range() function? Give one example.

Range Function Purpose — From Intuition to Precision

Imagine you have a machine that takes a number and spits out a result. You feed in 1, you get 3. You feed in 2, you get 5. You feed in 3, you get 7. The machine has a rule — maybe it doubles your input and adds 1. Now, the set of all numbers you could feed into the machine is called the domain. The set of all numbers that actually come out — the results you can get — is called the range.

That's the core idea. The range answers the question: "What are all the possible outputs of this function?"

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The Intuition First

Think of a function as a vending machine. You press a button (input), and a snack drops (output). The domain is the set of all buttons on the machine. The range is the set of all snacks that are actually inside — not the ones you wish were there, but the ones that can really come out.

If the machine has buttons A, B, C but only snacks chips and chocolate inside, then pressing any button gives you either chips or chocolate. The range is {chips, chocolate}. You can never get a sandwich, no matter which button you press.

For a mathematical function, the range is the same idea: the collection of every value the function can actually produce, given every allowed input.

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The Precise Statement

Let $f$ be a function from a set $A$ (the domain) to a set $B$ (the codomain). The range of $f$, often written as $f(A)$ or $\text{Range}(f)$, is the set of all outputs $f(x)$ where $x$ is in the domain $A$.

In set-builder notation:

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Why the Distinction Matters

Consider $f(x) = x^2$ with domain $\mathbb{R}$ (all real numbers) and codomain $\mathbb{R}$.

• The codomain is $\mathbb{R}$ — all real numbers.
• But $x^2$ is never negative. The range is $[0, \infty)$ — only non-negative numbers.

The codomain tells you what kind of numbers the function lives in; the range tells you what it actually hits.

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How to Find the Range

There is no single formula. You need to think about what the function can and cannot do.

1. For simple functions: Solve $y = f(x)$ for $x$ in terms of $y$. Then ask: for which $y$ does a valid $x$ exist in the domain? Those $y$ are the range.
2. For graphs: Look at the vertical extent of the graph — the set of $y$-coordinates the curve reaches.
3. For real-world functions: Consider physical constraints. If $f(t)$ is the height of a ball thrown upward, the range is from 0 to the maximum height — not all real numbers.

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A Worked Example

Find the range of $f(x) = \frac{1}{x-2}$ with domain $\mathbb{R} \setminus \{2\}$.

Set $y = \frac{1}{x-2}$. Solve for $x$:

For $x$ to be defined, we need $y \neq 0$ (division by zero). Also, $x$ can be any real number except 2 — and $2 + \frac{1}{y}$ can be any real number except 2 for $y \neq 0$. So the range is all real numbers except 0.

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The Bottom Line

The range is the set of actual outputs of a function. It is not the same as the codomain (the set of possible outputs by definition). To find it, you must examine what values the function can really produce, given its rule and its domain.