MathsMediumClass 11

Types of Functions — One-one, onto, bijective

Sets, Relations & Functions

JEE Qs

Hard

min

Always explicitly identify and carefully consider the domain and codomain of the function before testing for one-one or onto properties, as these sets are fundamental to the definitions and outcomes.

🧮 Key Formulas

A function f: A -> B is one-one (injective) if for all x1, x2 in A, f(x1) = f(x2) implies x1 = x2.

A function f: A -> B is onto (surjective) if for every y in B, there exists at least one x in A such that f(x) = y. This is equivalent to Range(f) = Codomain(f).

A function f: A -> B is bijective if it is both one-one (injective) and onto (surjective).

For finite sets A and B with |A|=m and |B|=n:

Number of one-one functions from A to B: P(n,m) = n! / (n-m)! if n >= m, else 0.

Number of onto functions from A to B: Sum[k=0 to n] ( (-1)^k * C(n,k) * (n-k)^m ) if m >= n, else 0.

Number of bijective functions from A to B: n! if m=n, else 0.

✅ Key Points for JEE

1The Horizontal Line Test: A function is one-one if and only if no horizontal line intersects its graph at more than one point.
2To prove a function is onto, show that its range is exactly equal to its codomain. This often involves solving y = f(x) for x in terms of y, then ensuring x exists in the domain for all y in the codomain.
3The domain and codomain are critical: Changing either can alter whether a function is one-one, onto, or bijective (e.g., f(x)=x^2: R->R is neither, but f(x)=x^2: [0,inf)->[0,inf) is bijective).
4For algebraic testing of one-one, assume f(x1)=f(x2) and rigorously deduce x1=x2. If you find x1= +/- x2 or other options, it's not one-one.
5When dealing with finite sets, specific combinatorial formulas are used to count the number of one-one, onto, or bijective functions.

⚠️ Common Mistakes

✕Confusing the codomain with the range; incorrectly assuming the codomain is always R (real numbers) if not explicitly stated, leading to errors in determining surjectivity.
✕Making algebraic errors when trying to solve f(x1)=f(x2) for x1=x2 or when finding the range by expressing x in terms of y.
✕Incorrectly applying the Horizontal Line Test, especially for piecewise functions or functions with restricted domains.
✕Forgetting to verify if the 'x' found when solving y=f(x) for surjectivity actually lies within the function's defined domain for every 'y' in the codomain.

NCERT Chapters

Class 11 Maths Ch 2: Relations and Functions