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Piecewise-defined Functions
A piecewise function uses different rules on stated intervals; choose the rule before substituting.
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Piecewise-defined Functions
What is it?
A piecewise function uses different rules on stated intervals; choose the rule before substituting.
In simple words
- A piecewise function uses different rules on stated intervals; choose the rule before substituting.
- Worked pattern: For f(x)=x+1 if x<0 and x2 if x>=0, f(-2)=-1 and f(3)=9.
- Final check: domain, sign, notation, units, and reasonableness.
Important terms
Piecewise-defined functions: A piecewise function uses different rules on stated intervals; choose the rule before substituting.
What you should learn
- State the rule for piecewise-defined functions.
- Solve a piecewise-defined functions problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
A piecewise function uses different rules on stated intervals; choose the rule before substituting.
CONCEPT AND RULE
A piecewise function uses different rules on stated intervals; choose the rule before substituting.
METHOD
For piecewise-defined 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
For f(x)=x+1 if x<0 and x2 if x>=0, f(-2)=-1 and f(3)=9.
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 piecewise-defined 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
- Piecewise-defined functions
- rule
- worked example
- worked example
How the ideas connect
Worked example: For f(x)=x+1 if x<0 and x2 if x>=0, f(-2)=-1 and f(3)=9.
Deeper understanding
Go beyond memorising Piecewise-defined Functions. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: For f(x)=x+1 if x<0 and x2 if x>=0, f(-2)=-1 and f(3)=9.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- Piecewise-defined functions
- rule
- worked example
- Piecewise-defined functions: A piecewise function uses different rules on stated intervals; choose the rule before substituting.
- CONCEPT AND RULE: A piecewise function uses different rules on stated intervals; choose the rule before substituting.
- METHOD: For piecewise-defined functions, identify the given information and required result. Select the matc
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 Piecewise-defined Functions: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Piecewise-defined functions
- rule
- worked example
- Piecewise-defined functions: A piecewise function uses different rules on stated intervals; choose the rule before substituting.
Important formulas
- Piecewise-defined 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.
Piecewise-defined functions
rule
worked example
Piecewise-defined functions: A piecewise function uses different rules on stated intervals; choose the rule before substituting.
A piecewise function uses different rules on stated intervals; choose the rule before substituting.
- Definition
- A piecewise function uses different rules on stated intervals; choose the rule before substituting.
- 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 Piecewise-defined Functions.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of Piecewise-defined Functions and say why it fits.
Mistake
Q4. State one common mistake students make about Piecewise-defined Functions.
Exam note
Q5. Write a short exam note on Piecewise-defined Functions: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.