Functions

What is a Function?

A function is a special kind of relation — the most important one in mathematics. You can think of it as a rule that takes an input and gives you exactly one output. If a relation is a set of ordered pairs, a function is that relation with two strict conditions:

1. Every element of the first set (the domain) must appear as a first element in some ordered pair. No element is left out.
2. No element of the domain can appear as the first element in two different ordered pairs with different second elements. Each input has exactly one output.

Definition 5 (from the textbook) states this formally: A relation $f$ from a set $A$ to a set $B$ is said to be a function if every element of set $A$ has one and only one image in set $B$.

In other words, a function $f$ is a relation from a non-empty set $A$ to a non-empty set $B$ such that:

• The domain of $f$ is $A$ (every element of $A$ is used).
• No two distinct ordered pairs in $f$ have the same first element (no input maps to two different outputs).

If $f$ is a function from $A$ to $B$ and $(a, b) \in f$, we write $f(a) = b$. Here:

• $b$ is called the image of $a$ under $f$.
• $a$ is called the preimage of $b$ under $f$.

The notation $f: A \to B$ means "$f$ is a function from $A$ to $B$".

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Checking Whether a Relation is a Function

Example 10

Let $N$ be the set of natural numbers. Define a relation $R$ on $N$ by

• Domain of $R$: $N$ (all natural numbers).
• Codomain of $R$: $N$ (the set in which we look for images).
• Range of $R$: The set of even natural numbers $\{2, 4, 6, 8, \dots\}$.

Is $R$ a function? Yes. Every natural number $n$ has exactly one image: $2n$. No natural number maps to two different values. So this relation satisfies both conditions of a function.

Example 11

Examine each relation and state, with reasons, whether it is a function.

(i) $R = \{(2,1), (3,1), (4,2)\}$

The domain elements are $2, 3, 4$. Each appears exactly once as a first element, and each has a unique image. This is a function.

(ii) $R = \{(2,2), (2,4), (3,3), (4,4)\}$

The first element $2$ appears twice, with images $2$ and $4$. This violates the "one and only one image" rule. This is not a function.

(iii) $R = \{(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)\}$

Every first element $1,2,3,4,5,6$ appears exactly once, each with a unique image. This is a function.

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Real Functions and Real-Valued Functions

Definition 6

A function whose range is either $\mathbb{R}$ (the set of real numbers) or a subset of $\mathbb{R}$ is called a real-valued function.

If, in addition, its domain is also either $\mathbb{R}$ or a subset of $\mathbb{R}$, it is called a real function.

In practice, most functions you study in Class 11 are real functions — they take real numbers as input and give real numbers as output.

Example 12 …