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

Question

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. 2 y = ——— x - 3

Accepted Answer

Hello. Today we're going to be determining whether the given relation between X and Y is a function, then we're going to be finding the domain and range. So we are given Y is equal to seven divided by the quantity X minus six. Now, if we use a graphing utility, the shape of the graph is going to be the graph of rational function where we have a vertical line or an Asymptote at X is equal to six, there is no X intercept for this rational function. So in order to determine whether this equation is a function or not, we're going to perform the, the vertical line test, we're going to draw a few vertical lines that intercept the graph. And as long as these vertical lines intercept the graph at exactly one point that will show that the given relation is a function. So the first vertical line intercepts the graph at exactly one point. And the same can be said for the 2nd, 3rd, 4th and 5th vertical line, since all of the vertical lines intercept the graph at exactly one point that shows that the given relation is a function. Now what we need to do is we need to determine the domain of the function. Now notice that this graph has two parts. The first part of the graph is to the left of the Asymptote. And the second part of the graph is to the right of the Asymptote. So that means we're going to have to union two parts of the domain. Now, the domain focuses on the lowest value of X to the highest value of X. The lowest value of X is going to be a negative infinity since the graph decreases to negative infinity. So the first part of the domain will be negative infinity. And the first part of the graph will approach the X value at six but never intercept the value of six. So that means the first part of the domain will end at positive six. But since six is in in to the graph, we're not going to include that value in the domain. Then we need to write the second part of the domain. Now, the second part of the domain starts at the value X is equal to six. So we're going to start up again at positive six. But since that value is not included, we're going to be assigned a parenthesis. Now notice that the second part will increase to positive infinity as the graph continues. What this means is that the largest value of X is going to be positive infinity. And the second portion of the domain will be from six to positive infinity. And finally, we're going to union both parts of the domain. So the full domain will be from negative infinity to positive six union, positive six to positive infinity. Now, what we need to do is we need to identify the range. Now, the range focuses on the lowest value of Y to the highest value of Y. And just like before, there are two parts to the range. Notice that there is no X intercept along the rational function. What this means is that we never cross the value Y is equal to zero. Or in other words, why is equal to zero is not included in the graph. The lowest value of Y is going to be negative infinity since the first portion of the graph decreases to negative infinity. So the range is going to start at negative infinity. And as we increase, we're going to approach Y is equal to zero but never cross the X axis. So the first portion of the range is going to end at zero but not include the value of zero. Now, the second part of the range is going to start slightly above zero and increase the positive affinity. So what this means is that the second value for the range is going to start at zero but not include zero and end at positive infinity. Finally, we're going to union both of these parts and the full range is going to be from negative infinity to zero union, zero to positive infinity. So just to summarize, we use the vertical line test to show that Y is equal to seven divided by X minus six is a function. The domain is from negative infinity to positive six union, positive six to positive infinity. And the range is from negative infinity to zero union, zero to positive infinity. And with that being said, the answer to this problem is going to be c so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. 2 y = ——— x - 3

Key Concepts

Definition of a Function

Domain of a Function

Range of a Function

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