### Video Transcript

Determine whether the function
represented by the following figure is even, odd, or neither even nor odd.

And then we have a graph of the
function shown. So letβs recall how to check the
parity of a function, how to check whether itβs even or odd. Well, the first thing we do is ask
ourselves, is the domain of this function centered at π₯ equals zero? We might recall that the domain of
a function is the set of possible inputs, the set of values of π₯, that we can
substitute into the function. And we can read that domain from
the graph.

Now we do need to be a little bit
careful because the graph doesnβt actually appear to be defined at π₯ equals
zero. In fact, the domain is the union of
the left-closed right-open interval from negative eight to zero and the left-open
right-closed interval from zero to eight. This is centered at π₯ equals
zero. Zero is exactly halfway through
this domain. And so we can now say yes to this
question. And we are able to move on to the
next part.

We can say that if π of negative
π₯ is equal to π of π₯, the function is even, and itβs odd if π of negative π₯ is
equal to negative π of π₯. Well, one way we can establish
whether either of these is true is to choose a value of π₯. For instance, letβs use the point
π₯ equals five. When π₯ is equal to five, the value
of our function, the π¦-value, is negative one. So π of five is negative one. And then this means that negative
π₯ must be negative five. And so we need to read the π¦-value
when π₯ is negative five. π of negative five is also
negative one. So it does look like this might be
an even function. But letβs check with another
value.

Letβs choose π₯ equals one. π of one is roughly equal to
negative 4.1. Then negative π₯ will be equal to
negative one. And once again, π of negative one
is roughly equal to negative 4.1. And so for the two values weβve
tried, π of negative π₯ is equal to π of π₯. But actually, if we look carefully,
we see that that function itself has reflectional symmetry about the π¦-axis. And so indeed, every value of π of
π₯ must be equal to every value of π of negative π₯. And so we can say that the function
itself must be even.