Question

An ________ function f is one for which f(-x) = f(x) for every x in the domain of f: an ______ function f is one for which f(-x) = -f(x) for every x in the domain.

Answer 1

Answers:

1. Even
2. Odd

Explanation:

1. Visually an even function is when the curve has reflectional symmetry over the y axis. In terms of notation, we would write f(-x) = f(x). For example, the parabola y = x^2 fits the description. The jump from -x to x means we apply the opposite to the x input, which therefore will apply the y axis reflection. For example, x = 5 jumps to x = -5.
2. An odd function has symmetry with respect to the origin. We can rotate the curve 180 degrees around the origin to have it match up with itself. The cubic curve y = x^3 is a good example of this. In terms of notation, f(-x) = -f(x) for all x in the domain. Notice how the two negatives apply two reflections (one over each axis). Combining an x axis reflection and y axis reflection, in either order, will generate a 180 degree rotation around the origin.