9 min readConcept sheet1-mark MCQ; 2 or 3-mark 'show that f is one-one'; part of a 5-mark bijection proof; also a JEE favourite

One-one (injective) functions

A function f: A → B is one-one when different inputs always give different outputs: f(x₁) = f(x₂) forces x₁ = x₂.

One-one (injective)f(x₁) = f(x₂) ⇒ x₁ = x₂, for all x₁, x₂ in A

Many-oneNot one-one: two different inputs share an output. Example: f(x) = x² on R, since f(−2) = f(2) = 4.

Horizontal line testf: R → R is one-one if every horizontal line meets its graph at most once.

Domain mattersx² is many-one on R but one-one on N, because the negative inputs are gone.

What one-one means

A function f: A → B gives every element of A exactly one image in B. f is one-one (injective) if no two different elements of A have the same image. In symbols: f(x₁) = f(x₂) ⇒ x₁ = x₂. Said the other way round: x₁ ≠ x₂ ⇒ f(x₁) ≠ f(x₂). A function that is not one-one is called many-one. Roll numbers in a class are one-one: each student has a different roll number.

How to prove or disprove one-one

1
Take any x₁, x₂ in the domain with f(x₁) = f(x₂).
2
Write out this equation using the rule for f, and simplify it.
3★
Show that it leads to x₁ = x₂. Then f is one-one.
4
To disprove, give two different inputs with the same output, e.g. f(−1) = f(1).

★ key step

Example 1: a linear function

Problem

Show that f: R → R, f(x) = 3x − 7 is one-one.

Working

Rule: f is one-one if f(x₁) = f(x₂) ⇒ x₁ = x₂.
Let f(x₁) = f(x₂). Then 3x₁ − 7 = 3x₂ − 7.
Add 7 to both sides: 3x₁ = 3x₂.
Divide by 3: x₁ = x₂.

Answer

f is one-one.

Example 2: the same rule on two domains

Problem

Check whether f(x) = x² + 3 is one-one (i) as f: R → R, (ii) as f: N → N.

Working

(i) f(−3) = 12 and f(3) = 12, but −3 ≠ 3.
So two different inputs share the output 12: f is not one-one on R.
(ii) Let f(x₁) = f(x₂) with x₁, x₂ ∈ N. Then x₁² + 3 = x₂² + 3, so x₁² = x₂².
So x₁² − x₂² = 0, that is (x₁ − x₂)(x₁ + x₂) = 0.
x₁ + x₂ > 0 for natural numbers, so x₁ − x₂ = 0, that is x₁ = x₂.

Answer

x² + 3 is not one-one on R, but it is one-one on N.

Example 3: a rational function (board style)

Problem

Show that f: R − {1} → R, f(x) = (x + 2)/(x − 1) is one-one.

Working

Rule: f(x₁) = f(x₂) must give x₁ = x₂.
Let (x₁ + 2)/(x₁ − 1) = (x₂ + 2)/(x₂ − 1).
Cross-multiply: (x₁ + 2)(x₂ − 1) = (x₂ + 2)(x₁ − 1).
Expand: x₁x₂ − x₁ + 2x₂ − 2 = x₁x₂ − x₂ + 2x₁ − 2.
Cancel x₁x₂ and −2 from both sides: −x₁ + 2x₂ = −x₂ + 2x₁.
So 3x₂ = 3x₁, that is x₁ = x₂.

Answer

f is one-one.

How to draw an arrow diagram

Draw two ovals side by side. Label the left one A (domain) and the right one B (codomain).
Write the elements of A inside the left oval and the elements of B inside the right oval.
Draw an arrow from each element of A to its image in B. Every element of A must have exactly one arrow.
One-one: no element of B receives two arrows. Write 'one-one' under the diagram.
Many-one: some element of B receives two or more arrows. Circle that element.

In real life

Aadhaar numbers work like a one-one function: each person gets a different 12-digit number, so a number points to exactly one person. Railway PNR numbers do the same for bookings.

Exam angle

1 marks: “Is f: R → R, f(x) = |x| one-one?” No. f(−2) = f(2) = 2, but −2 ≠ 2.

2 marks: “Show that f: R → R, f(x) = x³ + 5 is one-one.” Let f(x₁) = f(x₂). Then x₁³ + 5 = x₂³ + 5, so x₁³ = x₂³. Every real number has exactly one real cube root, so x₁ = x₂. Hence f is one-one.

3 marks: “Show that f: R − {3} → R, f(x) = (2x + 1)/(x − 3) is one-one.” Cross-multiply f(x₁) = f(x₂): (2x₁ + 1)(x₂ − 3) = (2x₂ + 1)(x₁ − 3). Expand and cancel 2x₁x₂ and −3: −6x₁ + x₂ = −6x₂ + x₁. So 7x₂ = 7x₁ and x₁ = x₂.

Remember it

One-one: no output is shared. Test it with f(x₁) = f(x₂) ⇒ x₁ = x₂.