MathematicsChapter 1: Functions and Graphs
Piecewise-Defined Functions
A piecewise function uses different rules on stated intervals; choose the rule before substituting.
Learning objectives
- State the rule for piecewise-defined functions.
- Solve a piecewise-defined functions problem step by step.
- Check the result, restrictions, notation, and units.
Definition
A piecewise function uses different rules on stated intervals; choose the rule before substituting.
Where / symbols
- 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 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
- 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.
Worked example
- Worked example:
- For f(x)=x+1 if x<0 and x2 if x>=0, f(-2)=-1 and f(3)=9.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Piecewise-defined functions
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Piecewise-Defined Functions
Sticky Notes · Formula and Worked Revision
Core Idea
- A piecewise function uses different rules on stated intervals; choose the rule before substituting.
Rule / Formula
- A piecewise function uses different rules on stated intervals; choose the rule before substituting.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For f(x)=x+1 if x<0 and x2 if x>=0, f(-2)=-1 and f(3)=9.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- 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.
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