A function is even when f(−x) = f(x). Its graph is a mirror image across the y-axis.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Odd functions
A function is odd when f(−x) = −f(x). Its graph looks the same after a half turn about the origin.
Where the names come from
xⁿ is even exactly when n is even, and odd exactly when n is odd. The names were borrowed from the exponents.
Most are neither
x² + x is neither. Even and odd are special cases, not a classification every function fits.
The two symmetries
Even means f(−x) = f(x), giving reflection symmetry about the y-axis. Odd means f(−x) = −f(x), giving rotational symmetry about the origin. Both are algebraic conditions with immediate visual meanings.
Test by substitution
Compute f(−x) and compare with f(x) and −f(x). If it matches neither, the function is neither even nor odd — which is the case for most functions, and is a legitimate answer.
Where the names come from
A polynomial with only even powers is even; with only odd powers, odd. Mixed powers give neither. The terminology is descriptive of exactly this, which makes it easier to remember.
Symmetry halves the work
Knowing a function is even means analysing half its domain and reflecting. In calculus, an odd function integrates to zero over a symmetric interval. Symmetry is a labour-saving observation, not a decorative one.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Priya Test x² + x
Priya checks whether f(x) = x² + x is even, odd, or neither.
- Step 1
She works out f(−x): (−x)² + (−x) = x² − x.
Step 4: Your Turn
Practice makes it stick.
The Test
Problem 1 of 2
Is f(x) = x⁴ even, odd, or neither? Answer 1 for even, 2 for odd, 0 for neither.
The Other One
Problem 2 of 2
Is f(x) = x³ − x even, odd, or neither? Answer 1 for even, 2 for odd, 0 for neither.
Even, Odd, or Neither
1 of 8
x². 1 for even, 2 for odd, 0 for neither.
2 of 8
x⁵. 1 for even, 2 for odd, 0 for neither.
3 of 8
x² + 3. 1 for even, 2 for odd, 0 for neither.
4 of 8
x³ + 1. 1 for even, 2 for odd, 0 for neither.
5 of 8
Sort each function by its symmetry.
Tap something to move it.
- Empty
- Empty
- Empty
6 of 8
If f is odd, what must f(0) be?
7 of 8
f is even and f(3) = 7. What is f(−3)?
8 of 8
f is odd and f(3) = 7. What is f(−3)?
Step 5: Quick Check
Show what you know.
Question 1 of 1
f is odd and f(2) = −5. What is f(−2)?
What You Learned
- Even means f(−x) = f(x), a mirror across the y-axis.
- Odd means f(−x) = −f(x), a half turn about the origin.
- Most functions are neither, and an odd function must pass through the origin.