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=2/(x-3)

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 and range. In this problem, we have Y is equal to 87 divided by X minus 29. We're given four answer choices. A option A and B are that it's not a function. And then they have different values for the domain and range and option C and D we have that it is a function and again, different options for the domain and range. And we're gonna come back to these answer choices as we work through this problem. So let's start with the first question is this a function we have Y is equal to 87 divided by X minus 29. Now, if we want to know if this is a function, we want to think about what happens for every single X value we put in, OK? For every single XL U we put into this function, we're gonna have 87 divided by that value minus 29. OK? So for every single X value we put in, we're gonna get only one corresponding Y value. OK? We can't put a particular X value in here and get two different answers for why? That's not possible. And you can imagine that if we have this situation on the, on a graph, OK. Is this a function that means we want to pass the vertical line test, right? And so for every X value that we draw a vertical line from, we only touch the graph in one place. It's exactly the same thing we're doing here, but we're looking at it from the equation. All right. So this is a function perfect. So we've done the first part of the problem and we can eliminate option A and B from the answer choices because they said that it wasn't a function. Now, we're left with looking at the domain and the range. Now for the domain, we want to find all possible X values that we can use in this function. Looking at our function, we can see that we have a restriction on X. If we have the denominator, which is X minus 29 equal to zero, then we'll be dividing by zero and our function will be undefined. OK. That means we cannot have this situation. And so X cannot equal 29. If X is equal to 29 we have the denominator equal to zero. And so we have that X cannot equal 29. And if we write this as our domain in interval notation, that means we have the interval from negative infinity up to 29. And we take the union of that with the interval from 29 up to infinity. OK. And we're using round brackets to indicate that we're not including what this interval is telling us really is that we can have any X value except for exactly 29. And we just break that up into two intervals around that value. OK. So we have our domain and looking at our answer traces, we can see that that's the same domain for option C or D. OK. So we're still left with C and D and now we need to move to our range. Now, what about our range? Yeah. Well, we can get positive values. OK? If X is greater than 29 the denominator will be positive, we have a positive value divided by a positive value that gives us a positive. We can also have negative values. OK? If the denominator is negative, we could get negative values. But one thing we can never get is this function equal to zero. OK. The num writer is 87. So the num writer will never be equal to zero. So we cannot have Y equals zero and Y equals zero is not possible. And so our range is gonna be all possible Y values except for zero. And so we're gonna write it really similarly to how we wrote the domain. We're gonna split this into two intervals around zero to indicate that we're not including zero. So we have negative infinity up to zero with a round bracket to indicate that we aren't including zero. And we take the union of that with the interval from zero to infinity. So we're including absolutely everything except at exactly zero. And so that is our range. If we compare this to our answer traces, we can see that this matches with answer choice C and so that is gonna be the correct answer here. We have a function with a domain that includes everything except X is equals to 29 a range that includes everything except Y is equal to 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=2/(x-3)

Key Concepts

Function Definition

Domain and Range

Vertical Line Test

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