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2.Functions – One-One & Onto
Class 11 • Mathematics • NCERT
Functions – One-One & Onto
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This lesson explains one-one (injective) and onto (surjective) functions as prescribed by NCERT Class 11. Students learn how to identify these functions using definitions, diagrams, and simple tests.
Lesson Objectives
- Understand one-one functions.
- Understand onto functions.
- Distinguish between one-one and onto.
- Solve exam-oriented questions.
1. One-One (Injective) Function
A function f : A → B is called one-one if different elements of A have different images in B.
f(x₁) = f(x₂) ⇒ x₁ = x₂
Example:
f(x) = 2x on ℝ is one-one.
2. Onto (Surjective) Function
A function f : A → B is called onto if every element of B has at least one pre-image in A.
Range of f = Codomain
Example:
f(x) = x³, f : ℝ → ℝ is onto.
3. Function That Is Not One-One
If two different inputs give the same output, the function is not one-one.
Example: f(x) = x² on ℝ (since f(2) = f(−2))
4. Function That Is Not Onto
If at least one element of the codomain has no pre-image, the function is not onto.
Example: f(x) = x², f : ℝ → ℝ (negative numbers have no pre-image)
5. One-One and Onto (Bijective)
A function that is both one-one and onto is called bijective.
Only bijective functions have inverse functions.
Practice Questions (NCERT)
- Define one-one function.
- Define onto function.
- Is f(x) = 3x + 2 one-one?
- Is f(x) = x² one-one on ℝ?
- Is f(x) = x³ onto ℝ?
- When is a function called bijective?
- Can a function be one-one but not onto?
- Can a function be onto but not one-one?
- Which functions have inverses?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- Different inputs give different outputs
- Every element of codomain has a pre-image
- Yes
- No
- Yes
- One-one and onto
- Yes
- Yes
- Bijective functions
- Yes
© Aviate Learning – Relations & Functions (NCERT Class 11)