MathematicsChapter 1: Functions and Graphs
Even and Odd Functions
f is even if f(-x)=f(x), odd if f(-x)=-f(x); otherwise it is neither.
Learning objectives
- State the rule for even and odd functions.
- Solve a even and odd functions problem step by step.
- Check the result, restrictions, notation, and units.
Definition
f is even if f(-x)=f(x), odd if f(-x)=-f(x); otherwise it is neither.
Where / symbols
- Even and odd functions
- f is even if f(-x)=f(x), odd if f(-x)=-f(x); otherwise it is neither.
CONCEPT AND RULE
f is even if f(-x)=f(x), odd if f(-x)=-f(x); otherwise it is neither.
METHOD
For even and odd 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
x^4+2 is even, while x^3-x is odd.
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 even and odd 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:
- x^4+2 is even, while x^3-x is odd.
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
- Even and odd functions
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Even and Odd Functions
Sticky Notes · Formula and Worked Revision
Core Idea
- f is even if f(-x)=f(x), odd if f(-x)=-f(x); otherwise it is neither.
Rule / Formula
- f is even if f(-x)=f(x), odd if f(-x)=-f(x); otherwise it is neither.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- x^4+2 is even, while x^3-x is odd.
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
- f is even if f(-x)=f(x), odd if f(-x)=-f(x); otherwise it is neither.
- Worked pattern: x^4+2 is even, while x^3-x is odd.
- Final check: domain, sign, notation, units, and reasonableness.
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