Determine whether each relation defines y as a function of x. Giv...

Question

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-√x

Accepted Answer

Hey, everyone in this problem, we're asked to find out if the given equation represents a function. And also to find the domain in range. The equation we're given is Y is equal to negative 77 multiplied by the square root of X. We're given four answer choices A through D and we're gonna come back to those as we work through this problem. Now rewriting our equation Y is equal to negative multiplied by the square root of X. And we wanna figure out whether this represents a function. Now we know that the square root of X is a function multiplying by a constant is not gonna change that. So we expect that this equation is gonna be a function. Let's try to show it more carefully. OK. In order to show this as a function, we want to look at what happens for every X value. OK? For every single X value we substitute into this equation, we're gonna take the square root of it, which is gonna give us one value and then we're gonna multiply it by negative 77. OK? And that's gonna give us one Y value. So for every X value we get only one corresponding why value. OK. And what this means is that this is a function. And by definition, this is gonna be a function and you can think about this with your vertical line test. OK? If you had a graph of this equation and you were asked if it's a function, you would draw a vertical line and that vertical line is gonna represent one particular X value. And you wanna see if it intersects that graph once twice or more. OK? You want it to only intersect once, which means that there is only one Y value associated with every X value. OK. So we're doing the exact same thing as a vertical line test here, but we're doing it with the equation algebraically. All right. So we know this is a function now we can eliminate options A and B. OK. Those two answer choices say that this is not a function. And so we know that we do not have either of those. Now, let's move on to the second part of this question starting with the domain. The domain recall is gonna be all possible X values that we can substitute into this equation. That will give us a result a that this function is defined for. Now we're taking the square root of X, we know that we cannot take the square root of a negative number. And so we need X to be greater than, or equal to zero, writing this in interval notation like the answer traces that we have zero up to infinity. OK. And we're using a square bracket on zero to indicate that we include that value is we have our domain from zero up to infinity including zero. And now we're gonna switch over to our range and recall that the range is gonna be all the possible Y values that we get using the domain for a function. So for our range, we substitute in any X value, we're gonna take the square that is gonna give us a positive number. And we know that that square root is gonna be a positive number. Then we multiply it by negative 77 a negative multiplied by a positive would always give us a negative value. And so our range is actually gonna be from negative infinity all the way up to zero. OK? All of those negative values. And we're gonna include zero, we're gonna draw a square bracket there and we include zero for the case where we have X is equal to zero. OK? If X is equal to zero, we have the square root of zero which gives us zero. We multiply that by negative 77 we get that Y is equal to zero. OK? So zero is included in the range. And now we have our domain in our range and we can see that the correct answer is gonna be option C A, this is a function that has a domain from zero up to infinity, including zero in a range from negative infinity up to including zero. Thanks everyone for watching. I hope this video helped see you in the next one.

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=-√x

Key Concepts

Function Definition

Domain and Range

Square Root Function

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