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=√(4x+1)

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. And the equation we're given is Y is equal to the square root of 69 X plus one, we're given four answer choices A through D and we're gonna come back to those as we work through this problem. So rewriting our equation, we have Y is equal to the square root of 69 X plus one. And we want to figure out if this is a function now for a function, what we need is that every X value we substitute into this equation gives us only one corresponding Y value. Mhm So let's think about this. OK? For every X value we substitute in, what are we gonna get? Oh, we're gonna multiply it by 69. We're gonna add one and then we're gonna take the square root. There's no way to get more than one Y value here and we have just the positive square root we don't have plus or minus. So there's no second solution coming from there. So for every X value there's gonna be only one corresponding Y value. And what that means is that this represents a function. And you can think about this if you were, if you think about if you had the graph of this function and you wanted to perform your vertical line test, OK. Every vertical line you draw is at one particular X value. And you're looking to see whether there's only one Y value, OK? Whether that vertical line intersects the function in only one place. And that's exactly what we're doing here. But with the equation, every X value we put in, we only get one Y value. All right. So we have a function and we can look at our answer traces and already eliminate option A and B because they both say that this is not a function. So we're left with C R D. Let's get back to the rest of the problem and sort out which one it should be. And we're gonna do that starting with the domain. Now, the domain of this function is gonna be all of the possible X values we can put into this function or the function will be defined. We have a square root and we know that that square root is only gonna be defined where whatever is under the square root is positive or equal to zero. And we cannot take the square root of a negative number and get a real value. And so we need 69 X plus one, everything under the square root to be greater than or equal to zero. We wanna write this as a restriction on X. So let's move one to the right hand side by subtracting, we get 69 is greater than 69 X. Sorry is greater than or equal to negative one. We divide by 69 to isolate X and we get that X must be greater than or equal to negative one divided by 69. OK. So we have this restriction on our X value in order for the function to be defined. And that is exactly what we want for our domain. So our domain is gonna go from negative one divided by 69 all the way up to positive infinity. OK. Those are all the X values that are greater than that restriction we found. And we've used a square bracket on negative one divided by 69 to indicate that we are including that value. And we can have that value when we have that X value, we get Y is equal to zero. OK. The square room is zero, we have our domain. So we can see that option C and option D both have the same domain. OK? So we can't eliminate either option at this point. We need to move to the range and the range is gonna be all the possible wine values we get using our domain for this function. Now we know that the square root is gonna give us positive values. So we know that our range is gonna be from zero up to infinity. What we need to decide is whether we include zero or not. Well, we just said we can have X is equal to negative one divided by 69. If we do that, we get the square root of zero, which is equal to zero. OK. So we can have a Y value of zero. And so we're gonna use a square bracket there to indicate that we include zero. So our range is from zero to infinity including zero. And that tells us that our correct answer choice is gonna be option D, OK. And option C, we do not include that end point. And so that is incorrect. All right. So this is a function. We have the domain from negative one divided by 69 up to infinity, including that left end point. And we have the range from zero to infinity again, including that left end point. Thanks everyone for watching. I hope this video helped see you in the next one.

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Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y=√(4x+1)

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