# Video: Determining the Domain of a Quotient of Functions Involving Square Roots

Find the domain of the function 𝑓(𝑥) = 6/√(−99 − 8𝑥).

01:50

### Video Transcript

Find the domain of the function 𝑓 of 𝑥 equals six over the square root of 99 minus eight 𝑥.

Remember, the domain of a function is the complete set of possible values of the independent variable. In other words, it’s the set of all possible 𝑥 values, which makes the function work and will output real 𝑦 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 it’s 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, let’s 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 𝑥, and to find where this is not equal to zero, let’s set it equal to zero and solve for 𝑥. We’ll begin by squaring both sides of this equation, and we get negative 99 minus eight 𝑥 equals zero. We add eight 𝑥 to both sides. And we see that negative 99 equals eight 𝑥, and then we divide through by eight. And we get 𝑥 equals negative 99 over eight. Of course, this is the value of 𝑥 such that the square root of negative 99 minus eight 𝑥 is equal to zero. And so, for our domain, we need 𝑥 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 𝑥 must be greater than or equal to zero. To solve for 𝑥, we add eight 𝑥 to both sides to get negative 99 is greater than or equal to eight 𝑥. And then, we divide through by eight. And so, 𝑥 must be less than or equal to negative 99 over eight. And we have the two criteria; 𝑥 cannot be equal to negative 99 over eight and 𝑥 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 ∞ to negative 99 over eight.