1.2 Domain and Range

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📖 Main Content

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What is a Function? (Quick Recap)

A function is a rule that takes an input value and produces exactly one output value.

Think of a function like a machine: you put a number in, and exactly one number comes out.

Functions are written in two ways:

• f(x) = 2x + 1 — this tells you the rule
• f : x ↦ 2x + 1 — this is read as "f maps x to 2x + 1"

For a mapping to be a function, every single input must lead to only one output. This is why:

• One-one mappings ✅ are functions
• Many-one mappings ✅ are functions
• One-many mappings ❌ are NOT functions (one input gives two or more outputs)

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🔑 Domain

The domain is the set of all allowed input values (the x-values you are allowed to put into the function).

The domain is always stated as part of the function definition. For example:

$f(x) = 2x - 1, \quad -1 \leq x \leq 3$

Here, the domain is $-1 \leq x \leq 3$. This means x can be any value from −1 to 3, including both endpoints.

Common domain notation:

• $x \in \mathbb{R}$ means x can be any real number (any number on the number line)
• $x > 0$ means x must be greater than zero
• $-1 \leq x \leq 3$ means x is between −1 and 3, inclusive
• $x \geq 2$ means x can be 2 or greater

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🔑 Range

The range is the set of all possible output values (the f(x) values that come out of the function for the given domain).

The range depends on:

1. The rule of the function
2. The domain (what x-values you are allowed to use)

How to find the range:

Step 1: Sketch the graph of the function over the given domain. Step 2: Look at the y-values (output values) that the graph reaches — the lowest point and the highest point. Step 3: Write the range using the same inequality notation as the domain.