For each graph, determine whether y is a function of x. Give t...

Question

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

Graph showing two relations on an xy-plane with arrows, illustrating function determination and domain-range analysis.

Accepted Answer

Hey, everyone here we are asked to identify if the given graph represents a function. And we are also asked to find the domain and range. Here we have four answer choice options. ABC and D, each of which state, whether this graph is of a function or not a function as well as the state, a domain and range in interval notation. So here to begin this problem, we will start by identifying if we have a graph of a function or not a function. And so we do this by utilizing the vertical line test. And so with the vertical line test, we can first draw a vertical line on the left side of the graph crossing through this orange decreasing line. And we see that our, the vertical line only passes through one point on the graph and therefore the vertical line test passes. And we see if we repeat this on the right side of the graph that we have the same result and the vertical line test passes. And now if we draw a vertical line through the y axis, we see we do pass through two different points. However, one point is a solid point. And the other point is a hollowed out point. And because of this hollow point, our vertical line test still it works. And so we do have a function because our graph does not fail at any point, the vertical line test. So again, because it does not fail the vertical line test, we do in fact have a function. And so now we can move on to identifying our domain and range. So beginning with our domain again, recalling the domain are the X values for our function. So we see looking from left to right, looking at the orange decreasing line, we see we go from negative infinity to zero. And we see that is not included because this line ends on a hollowed point. So zero is closed by parentheses and negative infinity is always excluded and therefore also closed by parentheses. And now for the second part of our domain, we can look at the second piece of our graph going from left to right. We see we have this solid point and 02 which extends to positive infinity. So we see for the second port portion, we have a domain from zero included to positive infinity. And now just summarizing our graph here, we see we have 10 that is not included and 10 that is included. So we can simplify our domain as simply from negative infinity to positive infinity. So now we can move on to finding our range And even though on our graph, again, we have this hollowed point, we still see that the graph or the function the curve still touches this point at Y is equal to four. Therefore, we can conclude that our range does include four. So we go from the lower point which is two, all the way to positive infinity. Here we see two is included since we have a solid point. So it is closed by a bracket and then we go to positive infinity which is always excluded and closed by a parentheses. So here we have found our domain and range and also identified that we do in fact have a function on our graph. And so we are left with answer choice C where again, we do have a function with a domain from negative infinity to positive infinity and a range from two included to positive infinity. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

For each graph, determine whether y is a function of x. Give the domain and range of each relation.

Key Concepts

Definition of a Function

Domain and Range

Vertical Line Test

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