Video Transcript
Determine the variation function 𝑣
of ℎ of 𝑓 of 𝑥, which is equal to two over 𝑥 squared plus three at 𝑥 equals
zero.
The question has asked us to find
the variation function of 𝑓 of 𝑥. So, let’s recall what a variation
function is. The variation function 𝑣 of ℎ of a
function 𝑓 of 𝑥 at 𝑥 equals 𝑎 is given by 𝑣 of ℎ is equal to 𝑓 of 𝑎 plus ℎ
minus 𝑓 of 𝑎. So, we need to apply this formula
to our function. We have that 𝑓 of 𝑥 is equal to
two over 𝑥 squared plus three. We have also been told that we need
to find the variation function at 𝑥 equals zero. Hence, 𝑎 is equal to zero. So, we have that 𝑣 of ℎ is equal
to 𝑓 of ℎ minus 𝑓 of zero. We can now substitute ℎ and zero
into 𝑓 of 𝑥. We have that 𝑣 of ℎ is equal to
two over ℎ squared plus three minus two over zero squared plus three.
The second of these fractions can
be simplified to two over three. In order to perform the
subtraction, we need to make sure the two fractions have the same denominator. We can do this by multiplying the
first fraction by three over three and the second fraction by ℎ squared plus three
over ℎ squared plus three. We are allowed to do this since
both of the fractions which we have introduced here are equal to one. After completing the two
multiplications, we have six over three lots of ℎ squared plus three minus two lots
of ℎ squared plus three over three lots of ℎ squared plus three. Our fractions now have a common
denominator of three lots of ℎ squared plus three, so we are ready to subtract.
We subtract the second numerator
from the first numerator to get six minus two lots of ℎ squared plus three over
three lots of ℎ squared plus three. Next, we can simplify the
numerator. Expanding the parentheses, we have
that the numerator is equal to six minus two ℎ squared minus six. The negative six will cancel out
the positive six, so we are left with just negative two ℎ squared in the
numerator. Here, we reach our solution, which
is that the variation function of 𝑓 of 𝑥, which is equal to two over 𝑥 squared
plus three, at 𝑥 equals zero is equal to negative two ℎ squared over three lots of
ℎ squared plus three.
