Functions: Domain, Codomain and Range Explained | NIMCET

Quick answer: A function $f : A \to B$ is a relation in which every element of A is paired with exactly one element of B. A is the domain, B the codomain, and the set of actual outputs $f(A)$ is the range, so range ⊆ codomain. To find the domain of a real function, remove values that make a denominator zero, an even root negative, or a logarithm's argument non-positive.

Functions are special relations; see types of relations for the general idea.

Function or not?

A relation from A to B is a function when:

every element of A has an image, and
no element of A has two different images.

Relation on A = {1, 2, 3} to B = {p, q}	Function?
$\{(1, p), (2, q), (3, p)\}$	Yes
$\{(1, p), (2, q)\}$	No, 3 has no image
$\{(1, p), (1, q), (2, p), (3, q)\}$	No, 1 has two images

Vertical line test: a curve is the graph of a function of x if every vertical line meets it at most once.

Number of functions from A to B with $n(A) = m$ and $n(B) = n$ is $n^m$ .

Finding the domain

Expression contains	Condition
$\frac{1}{g(x)}$	$g(x) \ne 0$
$\sqrt{g(x)}$	$g(x) \ge 0$
$\frac{1}{\sqrt{g(x)}}$	$g(x) > 0$
$\log g(x)$	$g(x) > 0$
$\sin^{-1}g(x)$ or $\cos^{-1}g(x)$	$-1 \le g(x) \le 1$

For sums and products of functions, the domain is the intersection of the individual domains.

Example: $f(x) = \sqrt{x - 1} + \frac{1}{x - 3}$ needs $x \ge 1$ and $x \ne 3$ , so the domain is $[1, 3) \cup (3, \infty)$ .

Finding the range

Common methods:

Write $y = f(x)$ , solve for x, and find the values of y for which x is real and in the domain.
Use known bounds, such as $-1 \le \sin x \le 1$ or $x^2 \ge 0$ .
Complete the square for quadratics.

Example: Range of $f(x) = \frac{x}{x + 1}$ , $x \ne -1$ .

Step 1: $y(x + 1) = x$ , so $x = \frac{y}{1 - y}$ .

Step 2: This is defined for every $y \ne 1$ . Range: $\mathbb{R} - \{1\}$ .

Standard real functions

Function	Rule	Domain	Range
Identity	$f(x) = x$	ℝ	ℝ
Constant	$f(x) = c$	ℝ	$\{c\}$
Modulus	$\lvert x\rvert$	ℝ	$[0, \infty)$
Greatest integer	$[x]$ , the largest integer ≤ x	ℝ	ℤ
Signum	$1$ if $x > 0$ , $0$ if $x = 0$ , $-1$ if $x < 0$	ℝ	$\{-1, 0, 1\}$
Reciprocal	$\frac{1}{x}$	$\mathbb{R} - \{0\}$	$\mathbb{R} - \{0\}$
Square root	$\sqrt{x}$	$[0, \infty)$	$[0, \infty)$
Exponential	$a^x$ , $a > 0$ , $a \ne 1$	ℝ	$(0, \infty)$
Logarithm	$\log_a x$	$(0, \infty)$	ℝ

Greatest integer function facts

$[2.7] = 2$ and $[-2.7] = -3$ .
$[x + n] = [x] + n$ for any integer $n$ .
$x - 1 < [x] \le x$ .
The fractional part $\{x\} = x - [x]$ lies in $[0, 1)$ .

Worked examples

Example 1: domain with a logarithm

Find the domain of $f(x) = \log(4 - x^2)$ .

$4 - x^2 > 0$ , so $-2 < x < 2$ . Domain: $(-2, 2)$ .

Example 2: range by bounds

Find the range of $f(x) = 3 + 2\sin x$ .

Since $-1 \le \sin x \le 1$ , the range is $[1, 5]$ .

Example 3: counting functions

How many functions are there from a 3-element set to a 2-element set?

$2^3 = 8$ .

Example 4: greatest integer equation

Solve $[x] = 3$ .

$3 \le x < 4$ , so $x \in [3, 4)$ .

Summary table

Idea	Key point
Function	Each input has exactly one output
Range vs codomain	Range ⊆ codomain
Domain rules	No zero denominators, no negative even roots, log argument > 0
Number of functions	$n^m$
$[x]$	Largest integer ≤ x

Exam tips

For domain questions, list every restriction and take the intersection.
For range questions, solving for x in terms of y is the most reliable method.
Be careful with negative numbers in $[x]$ : $[-0.5] = -1$ .

Next, read one-one, onto and composite functions, or browse the NIMCET Set Theory chapter.

Common mistakes

Treating the codomain as the range.
Forgetting $x \ne$ values that make the denominator zero after simplifying.
Writing $[-2.3] = -2$ instead of $-3$ .
Counting functions from A to B as $m^n$ instead of $n^m$ .

Frequently Asked Questions

What is the difference between codomain and range?

The codomain is the set B in f : A → B. The range is the set of outputs actually produced, which is a subset of the codomain.

How do you find the domain of a function?

Find all real x for which the expression is defined: no zero denominators, no negative values under even roots, and positive arguments for logarithms.

How many functions are there from A to B?

If n(A) = m and n(B) = n, there are nᵐ functions.

What is the greatest integer function?

[x] is the largest integer less than or equal to x. For example, [2.7] = 2 and [−2.7] = −3.

Is every relation a function?

No. A relation is a function only if every element of the domain has exactly one image.