Video Transcript
Is the function 𝑓 of 𝑥 equals 𝑥
to the fifth power times tan of six 𝑥 to the fourth power even, odd, or neither
even nor odd?
Let’s recall how we check the
parity of a function. The first thing we do is check the
domain of the function. We need that to be centered at 𝑥
equals zero. Then, if the answer is no, we can
say that the function is neither even nor odd without performing any further
tests. If the answer is yes, though, we
say that it will be even if it satisfies 𝑓 of negative 𝑥 equals 𝑓 of 𝑥. And it will be odd if it satisfies
𝑓 of negative 𝑥 equals negative 𝑓 of 𝑥. Then, of course, if it satisfies
neither of these, it will be neither even nor odd.
So let’s think about the domain of
our function. Our function is the product of two
functions. It’s the product of 𝑥 to the fifth
power and tan of six 𝑥 to the fourth power. And so the domain of 𝑓 of 𝑥 will
be the intersection of the domains of the respective parts of the function. Well, 𝑥 to the fifth power is a
polynomial, so its domain is the set of real numbers or the open interval from
negative ∞ to ∞. But what about the domain of the
trigonometric part? Well, it’s all real numbers, except
those that make cos of six 𝑥 equal to zero. But since the values of 𝑥 that
make cos of six 𝑥 equal to zero are symmetrical about the 𝑦-axis, then we can say
that the domain of tan of six 𝑥 to the fourth power must be centered at 𝑥 equals
zero.
Since both domains are centered at
𝑥 equals zero, then we can answer yes to this first question, and we’re able to
move on. We now see that it’s even if 𝑓 of
negative 𝑥 is equal to 𝑓 of 𝑥 and odd if it’s equal to negative 𝑓 of 𝑥. And so let’s evaluate 𝑓 of
negative 𝑥. To do so, we replace each instance
of 𝑥 in our original function with negative 𝑥. And we get 𝑓 of negative 𝑥 is
negative 𝑥 to the fifth power times tan of negative six 𝑥 to the fourth power. We’ll evaluate each part in
turn. Let’s begin with negative 𝑥 to the
fifth power. Since the exponent is odd, when we
multiply this out, we’re going to get a negative result. Negative 𝑥 to the fifth power is
as shown.
But what about the tan
function? Well, we can actually quote the
result that tan of 𝑥 is odd, meaning that tan of negative 𝑥 is equal to negative
tan of 𝑥 and, in turn, the tan of negative six 𝑥 is equal to negative tan of six
𝑥. But of course, we’re raising this
to the fourth power. We’re raising it to an even
exponent. And we know when we raise a
negative number to an even exponent, the result is positive. And so tan of negative six 𝑥 to
the fourth power is just tan of six 𝑥 to the fourth power.
And so 𝑓 of negative 𝑥 is
therefore equal to negative 𝑥 to the fifth power times tan of six 𝑥 to the fourth
power. So does this satisfy either of our
criteria, is it even or odd? Well, yes. If we look at it carefully, we see
it’s the same as negative 𝑓 of 𝑥. 𝑓 of negative 𝑥 is equal to
negative 𝑓 of 𝑥. And so the function must be
odd.
