Cogito
AP Precalculus · Chapter 2 · Lesson 2
Even and Odd Functions
Two kinds of symmetry, and the many functions with neither.
11 problems · about 20 minutes · AP Precalculus 1.8
What this lesson teaches
The student classifies a function as even, odd, or neither using f(−x).
- 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.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf is odd and f(2) = −5. What is f(−2)?
Answer 5
Why 5.
x². 1 for even, 2 for odd, 0 for neither.
Answer 1
Why Mirror across the y-axis.
x⁵. 1 for even, 2 for odd, 0 for neither.
Answer 2
Why Half turn about the origin.
x² + 3. 1 for even, 2 for odd, 0 for neither.
Answer 1
Why A constant is an even term.
Build It Up
The same ideas with more to keep track of.
3 problemsx³ + 1. 1 for even, 2 for odd, 0 for neither.
Answer 0
Why Mixing an odd power with a constant.
Sort each function by its symmetry.
Answer Even: x⁶, cos x · Odd: x⁷, sin x · Neither: x² + x
Why Cosine is the even one of the pair.
If f is odd, what must f(0) be?
Answer 0
Why f(0) must equal −f(0).
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsf is even and f(3) = 7. What is f(−3)?
Answer 7
Why Even means equal.
f is odd and f(3) = 7. What is f(−3)?
Answer -7
Why Odd means opposite.
The Test: Is f(x) = x⁴ even, odd, or neither? Answer 1 for even, 2 for odd, 0 for neither.
Answer 1
Why Even.
The Other One: Is f(x) = x³ − x even, odd, or neither? Answer 1 for even, 2 for odd, 0 for neither.
Answer 2
Why Odd.