Horizontal and Vertical Asymptotes

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Question

Horizontal and Vertical Asymptotes

Determine the domain and range of y = (√16―x²) / (x―2).

Accepted Answer

Welcome back, everyone. In this problem, we want to find the domain and the range of the following function, Y equals the square root of 25 minus X2d divided by X minus 3. Now, to help us figure out the domain and the range, let's, let's first focus on the domain, OK? And for our domain, in our quotient, there are two things that would affect it, the numerator and the denominator. Now, notice the numerator is a term with a square root, OK? So, in the numerator, The condition from the square root must be that the expression under the square root must be non-negative because if it becomes negative, then that means this is going to be an imaginary number. So from the numerator, 25 minus X squared must be greater than or equal to 0, which means that X must be greater than -5. But less than 5, OK, and thus this gives the interval the open bracket. 55 closed bracket. Now, if we think about the denominator. Then the only thing that would make or or that the only thing that would affect the domain from the denominator is if the denominator is undefined. And our denominator would be undefined if it equals 0. So our domain is fine once X minus 3 is not equal to 0. Thus, that means X cannot be equal to 3 because 3 minus 3 would be equal to 0 and make our function undefined. So now if we look at these two conditions, this tells us then that the domain. Is going to go from 5 to 3 or closed parentheses union with open parentheses 35 closed bracket. OK. This is going to be the domain of our function. Next, let's start to think about our functions range. Now for our range here. That is, what do we know about the possible why values? Well, if we think about it, OK, since the square root of 25 minus X2 is going to be positive for X in the inside the interval negative to 5 inclusive, the sign of Y depends on the sign of the denominator X minus 3. Or 1 less than 3. OK. Then the denominator is going to be negative. OK. And now for X greater than 3, OK, then the denominator is positive. The denominator is positive. Now, what does that mean for us? Well, no, that means as X approaches 3 from the left, OK? X approaches 3 from the left, then Y is going to tend to negative infinity. On the other hand, as x approaches 3 from the right, then Y is going to approach or Y tends to positive infinity. And additionally, at the end points X equals -5 and 5, the numerator becomes 0, so Y equals 0. Hence, that means the range. Is going to be open parentheses negative infinity infinity closed parentheses. Thus, this is the domain of our function, and this is the range of our function. Thanks a lot for watching everyone. I hope this video helped.

Horizontal and Vertical Asymptotes

Determine the domain and range of y = (√16―x²) / (x―2).

Key Concepts

Domain of a Function

Range of a Function

Asymptotes

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