Types of Functions: One-One and Onto

Classifying functions as one-one and/or onto is a recurring CBSE question, both as a 1-mark MCQ and as part of longer invertibility proofs. The definitions are short; the skill is applying them cleanly.

Definitions

A function f : A → B assigns each element of A (domain) exactly one element of B (codomain).

• One-one (injective): distinct inputs give distinct outputs. f(x₁) = f(x₂) ⇒ x₁ = x₂.
• Onto (surjective): every element of the codomain B is an image. Range = codomain.
• Bijective: both one-one and onto.

How to test

• One-one: assume f(x₁) = f(x₂), simplify to x₁ = x₂. OR use calculus — a strictly increasing/decreasing function is one-one.
• Onto: take arbitrary y ∈ B, solve y = f(x) for x ∈ A; if a valid x always exists, f is onto.

Examples

• f : ℝ → ℝ, f(x) = 2x + 3 is bijective (linear, non-constant).
• f : ℝ → ℝ, f(x) = x² is neither (f(−1)=f(1), not one-one; negatives unattained, not onto).
• f : ℕ → ℕ, f(x) = 2x is one-one but not onto (odd numbers missed).

Counting (finite sets)

For |A| = m, |B| = n: one-one functions exist only if m ≤ n; onto functions exist only if m ≥ n. A bijection needs m = n.

Exam Tricks & Tips

• 🎯 Any strictly monotonic (increasing or decreasing) function is automatically one-one — check the sign of f′(x).
• 🎯 To disprove one-one, find two inputs with equal output (e.g. x² at ±a).
• 🎯 To disprove onto, name one codomain value never hit (e.g. negative values for x²).
• 🎯 Changing the codomain to the range instantly makes any function onto.
• 🎯 Even-degree polynomials over ℝ are never onto ℝ; odd-degree ones are onto.
• ❌ Common mistake: checking onto against the domain instead of the codomain — onto is entirely about covering B.

Expected exam pattern

1-mark MCQ identifying the type, or a 3-mark "show f is one-one and onto" as a stepping stone to invertibility.

Quick recap

One-one = no two inputs share an output; onto = every codomain value is hit; bijective = both. Test one-one algebraically or via monotonicity, onto by solving y = f(x).

Cover the answer, recall, then check. 12 cards on one-one and onto functions.

Q1. Define a one-one (injective) function.
A1. f(x₁) = f(x₂) ⇒ x₁ = x₂; distinct inputs give distinct outputs.

Q2. Define an onto (surjective) function.
A2. Every element of the codomain is an image; range = codomain.

Q3. What is a bijective function?
A3. One that is both one-one and onto.

Q4. Quick calculus test for one-one?
A4. If f is strictly increasing or strictly decreasing (f′ keeps one sign), it is one-one.

Q5. Is f(x) = x² from ℝ→ℝ one-one? Onto?
A5. Neither. f(−1)=f(1) breaks one-one; negatives are unattained so not onto.

Q6. Is f(x) = 2x + 3 from ℝ→ℝ bijective?
A6. Yes — a non-constant linear map is one-one and onto.

Q7. f : ℕ→ℕ, f(x) = 2x. Classify it.
A7. One-one but not onto (odd numbers are never outputs).

Q8. For finite sets, one-one f : A→B requires what?
A8. |A| ≤ |B|.

Q9. For finite sets, onto f : A→B requires what?
A9. |A| ≥ |B|.

Q10. How do you make any function onto instantly?
A10. Restrict the codomain to the range.

Q11. How to disprove onto?
A11. Exhibit one codomain element that no input maps to.

Q12. Are even-degree polynomials onto ℝ?
A12. No — they have a maximum or minimum, so some ℝ values are unattained.

Think of a function as a machine: feed in an input, get exactly one output. But two deeper questions decide whether the machine can be reversed and whether it covers everything: does each output come from only one input (one-one), and does every possible output actually appear (onto)? These two ideas run through calculus, inverses and the entire chapter.

Definition: A function f : A → B assigns to each element of the domain A exactly one element of the codomain B. The set of outputs actually attained is the range, a subset of B.

One-one (injective)

Definition: f is one-one / injective if different inputs give different outputs: f(x₁) = f(x₂) ⟹ x₁ = x₂ (equivalently x₁ ≠ x₂ ⟹ f(x₁) ≠ f(x₂)).

• Algebraic test: assume f(x₁) = f(x₂) and derive x₁ = x₂.
• Graphical test (Horizontal Line Test): a function is one-one iff no horizontal line meets its graph more than once.
• Calculus test: if f is strictly increasing or strictly decreasing on its domain (f′ > 0 throughout, or f′ < 0 throughout), it is one-one.

Example: f(x) = 2x + 3 on ℝ is one-one (2x₁+3 = 2x₂+3 ⟹ x₁ = x₂). But f(x) = x² on ℝ is not one-one, since f(2) = f(−2) = 4.

Onto (surjective)

Definition: f : A → B is onto / surjective if every element of the codomain B is the image of at least one element of A — that is, range = codomain.

• Test: pick an arbitrary y ∈ B and solve y = f(x) for some x ∈ A. If a valid x always exists, f is onto.

Example: f : ℝ → ℝ, f(x) = 2x + 3 is onto (given y, take x = (y−3)/2 ∈ ℝ). But f : ℝ → ℝ, f(x) = x² is not onto, because negative numbers in the codomain are never outputs.

Bijective — the best of both

Definition: f is bijective if it is one-one and onto. Only bijections are invertible, which is why this classification matters so much for the "invertible functions" lesson.

Counting fact (finite sets): if |A| = |B| = n, the number of bijections A → B is n!. If a function from A to B (both size n) is one-one, it is automatically onto, and vice-versa — a hugely useful shortcut.

Worked example

Question: Show f : ℝ → ℝ defined by f(x) = x³ + 1 is a bijection.

Solution:
Step 1 — One-one: f(x₁) = f(x₂) ⟹ x₁³ + 1 = x₂³ + 1 ⟹ x₁³ = x₂³ ⟹ x₁ = x₂ (cube is one-one on ℝ). ✓
Step 2 — Onto: take any y ∈ ℝ. Solve y = x³ + 1 ⟹ x = (y − 1)^(1/3), which is a real number for every y. So a pre-image always exists. ✓
Conclusion: f is one-one and onto ⟹ bijective.

Exam application

Deciding injectivity/surjectivity is the gateway to inverses and to counting problems ("how many onto functions from a 3-set to a 2-set?" — answer 2³ − 2 = 6). In JEE, f′ sign analysis is the fastest injectivity check for messy functions.

Mnemonic: "One-one = no output shared; Onto = no output spared."

Key formulas / definitions

• One-one (injective): f(x₁) = f(x₂) ⇒ x₁ = x₂ (distinct inputs → distinct outputs).
• Onto (surjective): range = codomain (every y has a pre-image).
• Bijective: one-one and onto.
• Counting (|A| = m, |B| = n): total functions nᵐ; injective (n ≥ m) = n!/(n−m)!; bijective (m = n) = n!.

Test injectivity via the horizontal-line test; a bijection sets up a one-to-one correspondence between domain and codomain.

Worked Example

Problem: Determine whether the function f : R → R defined by f(x) = x² is one-one (injective) and onto (surjective).

Solution:
Step 1 — Test injectivity (one-one). A function is one-one if different inputs give different outputs. Consider f(2) = 4 and f(−2) = 4: two different inputs (2 and −2) give the same output.

Step 2 — Conclude injectivity. Since f(2) = f(−2) but 2 ≠ −2, the function is NOT one-one.

Step 3 — Test surjectivity (onto). A function is onto if every element of the codomain (here all of R) has a pre-image. The output x² is always ≥ 0.

Step 4 — Conclude surjectivity. Negative numbers (e.g. −4) in the codomain R have no real pre-image (no real x gives x² = −4). So the function is NOT onto.

Answer: f(x) = x² on R → R is neither one-one nor onto.