Lesson 6 of 6
Test opposite inputs and distinguish even, odd and neither.
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Big ideas · 1 of 9
Big ideas
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An even or odd function needs a domain symmetric about zero: if x is allowed, −x must also be allowed. A symmetric-looking formula on a restricted domain may fail this condition.
A function is even when for every domain input. Its graph reflects onto itself across the y-axis.
A function is odd when for every domain input. Its graph has origin symmetry, equivalent to a half-turn.
Substitute −x into the whole rule and simplify. One failed input pair disproves symmetry; a few matching pairs do not prove it for every input.
Step-through
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Worked example: test an even rule
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The function is neither even nor odd.
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Wrong idea: every function with an odd power is odd. An added nonzero constant can break origin symmetry. Test the full rule and its domain, not just its highest exponent.
Worked example
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Worked example
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The domain is symmetric, so g is odd.
Tip
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Class handout reference · Unit 2, section 2.9, pages 45-47. Use your class copy.
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Opposite inputs have equal outputs.
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What algebraic condition defines even symmetry?
on a symmetric domain.
What algebraic condition defines odd symmetry?
on a symmetric domain.
Which graph symmetry belongs to even functions?
Reflection across the y-axis.
Which graph symmetry belongs to odd functions?
A 180-degree rotation about the origin.
Is even on all real inputs?
Yes.
Is odd on all real inputs?
Yes.
Is on the domain even under the full domain definition?
No; its domain is not symmetric about zero.
Can a function be both even and odd?
Yes; the zero function on a symmetric domain is both.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Write it
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Match
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Connect the ideas
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Homework · 3 points
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Homework
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https://gradebridge-beryl.vercel.app/learn/precalculus/functions/even-and-odd-functions#homework
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