### Video Transcript

Find the domain of the function π
of π₯ equals six over the square root of 99 minus eight π₯.

Remember, the domain of a function
is the complete set of possible values of the independent variable. In other words, itβs the set of all
possible π₯ values, which makes the function work and will output real π¦
values. Now, we recall that there are two
main things that we need to consider. The denominator of any fractions
cannot be zero. And this is because when we divide
a number by zero, we say itβs undefined. We also say that the number under a
square root cannot be negative. It must be either greater than or
equal to zero. So, letβs consider the first
criteria, the denominator cannot be equal to zero.

The denominator of our fraction is
the square root of negative 99 minus eight π₯, and to find where this is not equal
to zero, letβs set it equal to zero and solve for π₯. Weβll begin by squaring both sides
of this equation, and we get negative 99 minus eight π₯ equals zero. We add eight π₯ to both sides. And we see that negative 99 equals
eight π₯, and then we divide through by eight. And we get π₯ equals negative 99
over eight. Of course, this is the value of π₯
such that the square root of negative 99 minus eight π₯ is equal to zero. And so, for our domain, we need π₯
cannot be equal to negative 99 over eight.

Now, in fact, the second criteria
says that the number under a square root cannot be negative. That is, negative 99 minus eight π₯
must be greater than or equal to zero. To solve for π₯, we add eight π₯ to
both sides to get negative 99 is greater than or equal to eight π₯. And then, we divide through by
eight. And so, π₯ must be less than or
equal to negative 99 over eight. And we have the two criteria; π₯
cannot be equal to negative 99 over eight and π₯ is less than or equal to negative
99 over eight. And of course, we want the
intersection of these. And so, we use these open brackets
when representing the interval. The domain of our function is the
open interval from negative β to negative 99 over eight.