Certain functions, such as rational and radical elementary functions, have instances of restricted domains. When finding the domain of a function, we must always remember that a rational function involves removing the values that could make the denominator of a fraction zero. Finding the domain of a function that is radical means not making the radical negative.
Of the seven parent functions, two of them aren't defined for all real numbers. They are the square root function and the reciprocal function. The square root function is only defined for values of x bigger than or equal to zero. So you would say that the domain is the set of real numbers bigger than or equal to zero and the way to describe that because it's a set is with with curly brackets, x such that x is greater than or equal to zero. Or you could do this in interval notation. You could write the interval from zero to infinity. Same set of numbers. Now I'll tend to in my in my lessons using a whole notation rather than set notation. Over here the reciprocal functions defined for all real numbers except x=0 so x doesn't equal zero. In set notation we'd say all x such that x doesn't equal zero or it's a little more cumbersome in interval notation that you'd say negative infinity to zero, union zero to infinity. So the negative numbers or the positive numbers. So this would be the domain of our reciprocal function. Now we create new functions from our parent functions and whenever we do, whenever we create functions with from these two, the domain maybe restricted and so let's see some examples of that. For example, this function, a radical function is only going to be defined when the inside function is greater than or equal to zero. So we will need 36-3x to be greater than or equal to zero. Which means 36 has to be greater than or equal to 3x and divide both sides by 3 you get x is less than or equal to 12, and so the domain will be all the numbers from 12 to the left on the number line. Now this function is a rational function which we'll study later. It's only going to be defined for real numbers other than 8 and -3. Right, because the denominator will be 0 there so x doesn't equal 8 or -3. And this is a little cumbersome in interval notation but you'd say from negative infinity to -3, 3, -3 to 8, and 8 to infinity. So these three intervals. And finally, here we have a kind of a a hybrid between radical functions and reciprocal function. Here x+9 has to be greater than or equal to zero. So in order for this part of the function to be defined. So x is going to be greater than or equal to -9 and then for this part to be defined, we can't have x=2. If I draw a little number line here. I'll put -9 right there, and I'll put 2 to the right. x greater than or equal to -9 refers to these numbers, right, here and forward but I have to avoid 2. So I have to put a little hole here. So let's just express this set in interval notation. We'd have the numbers between -9 and 2. So -9 to 2, not including 2, union the numbers from 2 to infinity. So that's how we find domain. We basically look out for two things, division by zero and the negative in the radical. We can't have either of those things and so as long as you start off by saying what's whatever is in the radical has to be greater than or equal to zero and whatever is in the denominator has to not equal to zero, then you'll be easily able to find the domain of a function.