AP® Precalculus · Unit 1 · Topic 1.5
Even and Odd Functions
A function is even if f(-x) = f(x), giving a graph mirrored across the y-axis, and odd if f(-x) = -f(x), giving a graph symmetric about the origin. Many functions are neither.
Even and odd describe how a graph behaves when you flip the input's sign. Plug in -x and see what comes out. If f(-x) simplifies back to f(x), nothing changed — the right side of the graph is a mirror image of the left side across the y-axis. That's why x2 looks like a mirror: (-x)2 = x2, so every point (a, a2) has a twin at (-a, a2) at the same height. If instead f(-x) simplifies to -f(x), the output flips sign along with the input. That's the pinwheel pattern of x3: (-x)3 = -x3, so the point at (a, a3) has a partner at (-a, -a3) — rotate the graph 180 degrees around the origin and it lands on itself.
The test is mechanical: substitute -x for every x, simplify, and compare to the original. Take f(x) = x5 - 3x. Then f(-x) = (-x)5 - 3(-x) = -x5 + 3x = -(x5 - 3x) = -f(x). That matches the odd pattern exactly, so f is odd. Every term here has odd degree (5 and 1), and that's not a coincidence — a single power term xn is even when n is even and odd when n is odd.
Trouble starts when a polynomial mixes degrees of different parity, like f(x) = x2 + x. Then f(-x) = x2 - x, which is neither x2 + x nor its negative -x2 - x. The even part and the odd part fight each other and neither symmetry survives. This matters for sketching and for understanding end behavior: even/odd symmetry is a shortcut, not a guarantee, and it sets up later work on function behavior where you can't always assume a tidy symmetric shape.
Key terms
- Even function
- A function where f(-x) = f(x) for every x, producing a graph symmetric across the y-axis.
- Odd function
- A function where f(-x) = -f(x) for every x, producing a graph symmetric about the origin.
- Origin symmetry
- A graph pattern where rotating the graph 180 degrees about the point (0,0) leaves it unchanged.
- Parity of degree
- Whether an exponent in a term is even or odd, which determines that single term's symmetry type.
Watch it explained
- How it worksHow does f(-x) = f(x) turn into reflection symmetry across the y-axis?
- How it worksWalk me through why f(-x) = -f(x) makes a graph symmetric about the origin.
- Worked exampleShow me whether f(x) = x⁵ - 3x is even, odd, or neither by testing f(-x).
- Common mistakeWhy isn't every polynomial either even or odd?
Check yourself
End of topic.