Even and Odd Functions
Even functions mirror across the y-axis; odd ones have origin symmetry.
About this topic
An even function satisfies f(−x) = f(x), so its graph is symmetric about the y-axis, like x² or cos x. An odd function satisfies f(−x) = −f(x), so it has rotational symmetry about the origin, like x³ or sin x.
To test, replace x with −x and compare. Many functions are neither. Recognising symmetry halves the work when graphing or integrating, and it simplifies many results in physics and calculus.
Worked example: test f(x) = x⁴ − 3x². Replace x with −x: (−x)⁴ − 3(−x)² = x⁴ − 3x², the same as f(x), so f is even. Now test g(x) = x³ + x: g(−x) = −x³ − x = −g(x), so g is odd. But h(x) = x² + x gives x² − x, which is neither f(x) nor −f(x), so h is neither. Even functions have only even powers of x, and odd functions only odd powers, as a handy shortcut.
Symmetry matters in physics, where an odd force law reverses direction under reflection, and in calculus, where the area under an odd function over a balanced interval such as −3 to 3 is exactly zero. Do not assume a function must be one or the other: most are neither. And remember f(x) = 0 is the only function both even and odd. Also, adding a constant to an odd function destroys its odd symmetry.
Key ideas
- Even: f(−x) = f(x)
- Odd: f(−x) = −f(x)
- x² is even, x³ is odd
Quick check
Use cases: where Even and Odd Functions helps
- Homework: review the key idea behind Even and Odd Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Even and Odd Functions without notes.
- Visual learning: play with the interactive lab to see how Even and Odd Functions changes when you adjust its inputs.
- Teaching a friend: explain Even and Odd Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Even and Odd Functions?
Even functions mirror across the y-axis; odd ones have origin symmetry.
What will I learn in Even and Odd Functions?
Even: f(−x) = f(x). Odd: f(−x) = −f(x). x² is even, x³ is odd.
Is Even and Odd Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Even and Odd Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Even and Odd Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.