What Is a Function in Maths? Definition, Examples & Function vs Relation

Every relation is not a function, but every function is a relation. This one line summarises one of the most important ideas in Class 11 Maths, and it’s where students often lose easy marks — mixing up the two terms or forgetting the one condition that separates them. Once you understand relations, domain, range and codomain , functions become a natural extension. This post lays out the exact definition, shows how to test whether a relation qualifies as a function, and introduces the real function.

Defining a Function

A relation f from a set A to a set B is called a function if every element of set A has one, and only one, image in set B. This “one and only one” condition is what makes a function special.

A function f is a relation from a non-empty set A to a non-empty set B such that:

1. The domain of f is the entire set A (every element of A must be used).

2. No two distinct ordered pairs in f share the same first element (each input maps to exactly one output).

A function from A to B is written as f: A → B. If (a, b) belongs to f, we write f(a) = b, where b is called the image of a under f, and a is called the preimage of b under f.

The Two Ways a Relation Can Fail to Be a Function

Understanding what makes a function work is easier once you see the two specific ways a relation can fail the test.

Failure 1: An Element of A Has No Image

If even one element of domain set A is left unconnected — no arrow pointing anywhere in B — the relation is disqualified. Every element of A must map to something.

Failure 2: An Element of A Has More Than One Image

If any element of A connects to two or more elements of B, the relation fails immediately. A function demands a single, unambiguous output for every input.

Worked Examples: Function or Not?

Function Invertibility & Bijection Sandbox

Adjust the slope and intercept of linear function f(x) = ax + b. Observe how a non-zero slope guarantees bijection and an inverse, whereas slope a = 0 collapses the function into a constant line with no inverse.

Function f(x) = ax + b

Slope a2

Intercept b1

Input x3

Output f(x)7

Inverse f⁻¹(x) 1

Inverse Exists? ^⚠ Yes

Function g(x) = cx + d

Slope c1

Intercept d-2

Input x3

Output g(x)1

Compositions

(f ∘ g)(x) = f(g(x)) 3

(g ∘ f)(x) = g(f(x)) 5

Is Commutative? (f ∘ g = g ∘ f) No ✗

⚠ Syllabus Checkpoints & Exam Watch-Outs

• ⚠ When slope a = 0, f(x) becomes a constant horizontal function. It is many-one, failing injectivity, so its inverse does not exist.

Testing a relation is a matter of checking these two conditions carefully.

• R = {(2,1), (3,1), (4,2)} — This is a function. Every element in the domain (2, 3, 4) has exactly one image, even though 2 and 3 happen to share the same image, 1. Repeated outputs are allowed; repeated inputs are not.

• R = {(2,2), (2,4), (3,3), (4,4)} — This is not a function. The element 2 appears twice as a first element, once mapped to 2 and once to 4. A single input cannot have two outputs.

• R = {(1,2), (2,3), (3,4), (4,5), (5,6), (6,7)} — This is a function. Every element has one and only one image, and no first element repeats.

A relation where the domain does not equal the full starting set also fails to be a function — for example, if A = {1,2,3,4,5,6} and R = {(1,2), (2,3), (3,4), (4,5), (5,6)}, the element 6 has no image at all, so R is a relation but not a function on A.

Real Valued Functions and Real Functions

Two closely related terms often appear together in this chapter, and it is worth keeping them separate.

• A real valued function is a function whose range is either the set of real numbers R, or some subset of R.

• A real function is a real valued function whose domain is also R or a subset of R.

In practice, most functions studied at this stage — polynomial, rational, modulus, and greatest integer functions — are real functions, since both their inputs and outputs are real numbers.

Completing a Function Table

A very common exercise is building a table of values once a function’s rule is known. For instance, take f: N → N defined by f(x) = 2x + 1. To complete a table for x = 1 to 7, you substitute each value of x into the rule one at a time:

f(1) = 2(1) + 1 = 3 f(2) = 2(2) + 1 = 5 f(3) = 2(3) + 1 = 7 f(4) = 2(4) + 1 = 9

The pattern continues predictably, giving f(5) = 11, f(6) = 13, and f(7) = 15. This kind of substitution practice builds the fluency needed for the algebra of functions covered later in the chapter.

Why the “One and Only One” Rule Matters

This condition is not just a technicality — it is the entire reason functions are so useful in mathematics, science, and even everyday planning. A function guarantees predictability: given any specific input, you always get exactly one specific, well-defined output. This reliability is what allows functions to model real situations — temperature conversion, distance versus time, or price versus quantity — where a single cause should always produce a single, consistent effect.

Quick Recap

• A function is a relation where every element of A has exactly one image in B.

• Two failure conditions: an element of A with no image, or an element of A with more than one image.

• f: A → B, with f(a) = b meaning b is the image and a is the preimage.

• A real function has both its domain and range as R or a subset of R.

Related Reading

• Relations in Maths: Domain, Range, Codomain & Arrow Diagrams

• Types of Functions: Identity, Constant, Polynomial & Rational Functions

• Algebra of Real Functions: Domain, Range & Solved Examples

Practice What You’ve Learned

Telling a function apart from a plain relation is one of the highest-yield skills in this chapter, and it only becomes second nature with repeated practice on varied examples. ChampionsPrep offers structured practice sets, instant feedback, and mock tests for CBSE and State Board Class 11–12 Commerce students, along with CUET, IPMAT, and NPAT aspirants. Registration is free, and you pay only as you use the platform.

Test Your Knowledge

Q1.A relation f from set A to set B is called a function if:

Some elements of set A have multiple images in set BEvery element of set B is mapped from at least two elements in set AEvery element of set A has one and only one image in set BThe range of f is equal to the empty set

Q2.In the vertical line test for functions, a curve in the Cartesian plane represents a function y = f(x) if:

Every horizontal line intersects the curve at least twiceThe curve passes through the origin (0, 0)The slope of the curve is positive everywhereNo vertical line intersects the curve at more than one point

Q3.Let A = {1, 2, 3} and B = {4, 5}. Which of the following relations is a valid function from A to B?

f = {(1, 4), (2, 5), (3, 4)}f = {(1, 4), (2, 5)}f = {(1, 4), (1, 5), (2, 4), (3, 5)}f = {(4, 1), (5, 2)}