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=√(7-2x)

Accepted Answer

Hey, everyone in this problem, we're asked to find out if the given equation represents a function and also to find its domain and range. The equation we're given is Y is equal to the square root of 13 minus 37 X. And we're given four answer choices A through D. And we're gonna come back to those as we work through this problem. Now, starting with writing out our equation, Y is equal to the square root of 13 minus 37 X. And we want to figure out if this is a function. Now remember that a function is defined OK? If every X value that you put into that function gives you only one corresponding value. OK? So let's think about this. If we were to put any X value in here, we would get 13 minus 37 multiplied by that X value. And we take the square root, we're only gonna get one part possible Y value. OK. For every X value for every X value, there is only one corresponding high value which tells us that this is a function. And important to know here, we have this square root. We're not writing plus or minus the square root. OK? So we do only have one Y value. If we were to write plus or minus, we would get two different Y values and that would be a different situation. But in this case, we're taking that positive route of minus 37 X, that is gonna give us only one Y value for every X value. So this represents a function. Now, we can eliminate options, A and B already. OK. A and B both say that this is not a function. OK? So eliminate those options. We're left with C and D. Now let's move to the domain and for the domain, we wanna look for all possible X values that we can substitute into this function. Now, we have a square root here and recall that we cannot take the square root of a negative number. What that means is that everything under the square root minus 37 X must be greater than our greater than, or equal to zero. If we have 13 minus 37 X greater than, or equal to zero, it tells us that 13 must be greater than, or equal to 37 X. And we add that 37 X to move it over. We divide both sides by 37 we get that X must be less than or equal to 13 divided by 37. So now that we've written this restriction on our X value, OK? We can write our domain and interval notation. We know that X must be less than, or equal to 13 divided by 37. And so we can write this as the interval. OK. If X is less than we can have X negative infinity all the way up to 13 divided by 37. And we're gonna put a square bracket on this right hand end to indicate that we can include that 0.13 divided by 37. OK. If we have X is equal to 13, divided by 37 we get Y is equal to the square root of zero, which is equal to zero. OK. So we can have that situation. And so that's gonna be the domain of this function. Now let's switch over to the range. OK? And the range is gonna be all possible Y values that we get when we use the X values in our domain. OK? We know that the square root is always gonna give us a positive answer. OK? It's gonna give us zero or something positive. And so the range is gonna be from zero up to infinity or we have a square bracket on zero to indicate that we can have a Y value of zero. OK? And we just showed how we can get that Y value and it's by using 13 divided by 37 as I, all right. So we have our domain and our range, we've already shown that this is a function. So we're looking at option C or D and the correct answer here is gonna be option D. OK? Option D says that this is a function. The domain is from negative infinity up to and including 13 divided by 37. And the range goes from zero up to infinity where we include zero, which is exactly what we've just found. 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=√(7-2x)

Key Concepts

Function Definition

Domain and Range

Square Root Function

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