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One-one and Many-one Functions
A one-one function gives different outputs for different inputs and passes the horizontal-line test.
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One-one and Many-one Functions
What is it?
A one-one function gives different outputs for different inputs and passes the horizontal-line test.
In simple words
- A one-one function gives different outputs for different inputs and passes the horizontal-line test.
- Worked pattern: f(x)=2x+1 is one-one; f(x)=x^2 on R is many-one.
- Final check: domain, sign, notation, units, and reasonableness.
Important terms
One-one and many-one functions: A one-one function gives different outputs for different inputs and passes the horizontal-line test.
What you should learn
- State the rule for one-one and many-one functions.
- Solve a one-one and many-one functions problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
A one-one function gives different outputs for different inputs and passes the horizontal-line test.
CONCEPT AND RULE
A one-one function gives different outputs for different inputs and passes the horizontal-line test.
METHOD
For one-one and many-one functions, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
f(x)=2x+1 is one-one; f(x)=x2 on R is many-one.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve one-one and many-one functions
How it works
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Remember
- One-one and many-one functions
- rule
- worked example
- worked example
How the ideas connect
Worked example: f(x)=2x+1 is one-one; f(x)=x^2 on R is many-one.
Deeper understanding
Go beyond memorising One-one and Many-one Functions. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: f(x)=2x+1 is one-one; f(x)=x^2 on R is many-one.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- One-one and many-one functions
- rule
- worked example
- One-one and many-one functions: A one-one function gives different outputs for different inputs and passes the horizontal-line test.
- CONCEPT AND RULE: A one-one function gives different outputs for different inputs and passes the horizontal-line test.
- METHOD: For one-one and many-one functions, identify the given information and required result. Select the m
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
For long answers on One-one and Many-one Functions: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- One-one and many-one functions
- rule
- worked example
- One-one and many-one functions: A one-one function gives different outputs for different inputs and passes the horizontal-line test.
Important formulas
- One-one and many-one functions
- rule
- worked example
Common exam traps
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- A matching-looking formula is not valid until its conditions and domain have been checked.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- Mistake
A matching-looking formula is not valid until its conditions and domain have been checked.
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
One-one and many-one functions
rule
worked example
One-one and many-one functions: A one-one function gives different outputs for different inputs and passes the horizontal-line test.
A one-one function gives different outputs for different inputs and passes the horizontal-line test.
- Definition
- A one-one function gives different outputs for different inputs and passes the horizontal-line test.
- Critical warning
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all d
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Exam Practice
Define
Q1. Define One-one and Many-one Functions.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of One-one and Many-one Functions and say why it fits.
Mistake
Q4. State one common mistake students make about One-one and Many-one Functions.
Exam note
Q5. Write a short exam note on One-one and Many-one Functions: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.