Piecewise, Even, and Odd Functions

One function, several rules

A piecewise function uses different formulas on different parts of its domain. Think of a phone plan: free under 100 minutes, then a per-minute charge above that. Each line of the definition carries a condition telling you which inputs it governs, and the conditions must not overlap.

          | x + 1,    if x < 0
f(x) =    | x^2,      if 0 <= x <= 2
          | 5,        if x > 2

f(-3) : -3 < 0,        use x + 1  -> -3 + 1 = -2
f(0)  : 0 in [0, 2],   use x^2    -> 0^2    = 0
f(2)  : 2 in [0, 2],   use x^2    -> 2^2    = 4
f(7)  : 7 > 2,         use 5      -> 5

Even functions: mirror across the y-axis

A function is even when f(−x) = f(x) for every x. Plugging in the opposite input gives the same output, so the graph is a mirror image across the y-axis. The model is f(x) = x^2; powers like x^4 and x^6, and the cosine, are even.

f(x) = x^2 - 4
  f(-x) = (-x)^2 - 4 = x^2 - 4 = f(x)   ->  EVEN

g(x) = x^4 + 3x^2 + 1
  g(-x) = x^4 + 3x^2 + 1 = g(x)          ->  EVEN

Odd functions, and the test that settles it

A function is odd when f(−x) = −f(x): the opposite input gives the opposite output. The graph has rotational symmetry about the origin — spin it 180° and it lands on itself. The model is f(x) = x^3; odd powers and the sine are odd. Most functions are neither, and that is a perfectly good answer.

1. Compute f(−x) by replacing every x with −x, then simplify.
2. If f(−x) = f(x), it is even. If f(−x) = −f(x), it is odd.
3. If it is neither, say so. Example: f(x) = x^3 + 1 gives f(−x) = −x^3 + 1, which matches neither f(x) nor −f(x) — neither.