Even and odd functions are functions that satisfy certain properties. This is a powerful concept; identifying even and odd functions can make some seemingly tough integration problems trivial.
Even functions are functions that satisfy for all . Even functions are symmetric about the line .
Odd functions are functions that satisfy for all . Odd functions exhibit point symmetry about the origin.
It is possible for a function to be neither odd nor even.
The function is
even but not odd
odd but not even
both even and odd
neither even nor odd
If we evaluate , we get:
Therefore, is even and obviously not odd, so the answer is choice .
Even and odd functions can also be spotted by their graphs - specifically, even functions have the y-axis as a line of symmetry and odd functions have rotational symmetry about the origin. For example, take a point (a,b) on the coordinate plane such that the function f(x) is even, thus f(x) = f(-x) - so if (a,b) is on the graph, (-a,b) is as well. Odd functions can be rotated 180 degrees about the origin and then appear the same, and so are said to be symmetric about the origin.