Video Transcript
Find the domain of the function 𝑓
of 𝑥 equals six over the square root of nine minus 25𝑥 squared in the set of real
numbers.
Remember, the domain of a function
is the complete set of possible values of the independent variable. Here, that’s 𝑥. We essentially say that it is the
set of all possible 𝑥 values, which makes the function work and outputs real 𝑦
values. When finding the domain, we need to
consider that the denominator of some fraction or rational function can’t be equal
to zero. That’s because when we divide a
real number by zero, we say that we get a result that’s undefined. We also know that the number
underneath a square root sign cannot be negative, absolutely must be positive. So, let’s consider how this impacts
on our function.
We have a rational function, so the
denominator, the square root of nine minus 25𝑥 squared, cannot be equal to
zero. So, let’s assume it is equal to
zero and solve for 𝑥. They will tell us the values of 𝑥
that we cannot use. We begin by squaring both sides of
this equation. And we get nine minus 25𝑥 squared
equals zero. We then factor nine minus 25𝑥
squared, and we could use a difference of two squares. We get three minus five 𝑥 times
three plus five 𝑥. Now, for the product of these two
expressions to itself be equal to zero, either three minus five 𝑥 must be zero or
three plus five 𝑥 must be zero. Let’s solve the first equation for
𝑥. We’re gonna add five 𝑥 to both
sides. And we get three equals five
𝑥. We then divide through by five, and
we get 𝑥 equals three-fifths.
To solve the second equation, we
subtract three from both sides to get five 𝑥 equals negative three. We then divide through by five, and
we found the second solution to the equation. It’s 𝑥 equals negative
three-fifths. Now of course, these are the values
for which the denominator of the fraction is equal to zero. So, we say that part of our domain
is the fact that 𝑥 cannot be equal to either positive or negative three-fifths.
We’ll now consider the second part,
the number inside the square root must itself be positive. That is, nine minus 25𝑥 squared is
greater than zero. Now, we just solved the equation
nine minus 25𝑥 squared is equal to zero. And we got 𝑥 equals three-fifths
and 𝑥 equals negative three-fifths. The graph of 𝑦 equals nine minus
25𝑥 squared is a parabola. It’s an inverted parabola since the
coefficient of 𝑥 squared is negative. And it has 𝑥 intercepts at
three-fifths and negative three-fifths.
Now, the part of the curve that is
greater than zero and therefore nine minus 25𝑥 squared is greater than zero is the
bit that I’ve highlighted pink. These are all values of 𝑥 in the
open interval negative three-fifths to three-fifths. In other words, values of 𝑥
greater than negative three-fifths and less than three-fifths. Now, to find the domain of our
function, we’re looking for the intersection of these two results. Now, in fact, values of 𝑥 in the
open interval negative three-fifths to three-fifths are also not equal to positive
or negative three-fifths. And so, the intersection of our
results and therefore the domain of our function is the open interval from negative
three-fifths to three-fifths.
