Video Transcript
Is the function represented by the
figure even, odd, or neither even nor odd?
Remember, we can check the parity
of a function by considering its graph. Before we do, though, we do need to
decide whether the domain is centered at 𝑥 equals zero. If the answer is yes, then we move
on to the next stage. But if it’s not, the function
cannot be even nor odd. So let’s find the domain of our
function. We remember that the domain is the
set of possible inputs to the function, the set of values of 𝑥 that go into the
function. The smallest value of 𝑥 that is in
our function is 𝑥 equals two, and the largest possible value of 𝑥 is 𝑥 equals
six. And so we see that the domain is
the closed interval from two to six. This domain is actually centered at
𝑥 equals four. The halfway point is 𝑥 equals
four. And so actually, we’re answering no
to this first question. And so the function is neither even
nor odd.
But let’s have a look at a common
misconception here. When we think about the graphical
representation of functions, we know that these functions are even if they have
reflectional symmetry about the 𝑦-axis. And they’re odd if they have
rotational symmetry order two about the origin. Now, our graph does appear to have
some rotational symmetry. If we take the center to be four,
one, then it does indeed have rotational symmetry order two. If I rotate that graph 180 degrees,
it will look exactly the same. But this is not about the
origin. Its center is four, one. And so that confirms to us that
this graph is neither even nor odd.
