Determine whether each relation defines a function, and give t...

Question

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

Graph of a function showing a curve with peaks and valleys, illustrating function behavior.

Accepted Answer

Hey, everyone in this problem, we're asked to identify if the given graph represents a function and also define the domain in range. Now, the graph we're given looks kind of like a cubic function. It starts in the upper left. In the second quadrant, it decreases, it comes down to a minimum, a local minimum. It increases again to a local maximum around X equals five and then decreases down into quadrant four. We're given four answer choices. Option A, it's not a function. The domain is from zero to infinity. Noninclusive. And the range is from negative infinity to infinity. Option B, it's not a function and the domain and range are both negative infinity to positive infinity. Option C it is a function. The domain is from zero to infinity, not including the end points. And the range is from negative infinity to positive infinity. And option D, it is a function and both the domain and range are from negative infinity to positive infinity. Now, when we're given a graph and we're asked whether or not it's a function, what we wanna do is perform our vertical wine test. So to perform our vertical line test what we're gonna do is we're gonna go to our graph and we're gonna draw vertical lines anywhere on this graph. So you draw a vertical line anywhere on this graph, right? And that vertical line is only gonna touch that graph in one point every time we draw a vertical line, it only touches the curve in one point. That means that it passes the vertical line test and it is a function. OK? What this tells us is that each X value only has one corresponding Y value. OK? Which is the definition of a function. And so our vertical line test that gets a green check mark that is passed. OK, which tells us that this is a function. So we've already narrowed down our answer choices. We know we can eliminate option A and option B because those say that it's not a function, it is a function. Now let's do the domain and range. So what about the domain? Well, the given the graph, OK, we're showing that this function has arrows at the end points. It's going off to the far left, it's going off to the far right. We aren't showing any restrictions at all. And so the domain is gonna be all real numbers. We have negative infinity to positive infinity for the range. If we look at the end points on the left end point, this function is going up, OK? And on the right hand side, it's going down and So the range is gonna stretch from all of those numbers up in positive infinity, all the way down to negative infinity. And so the range as well is going to be from negative infinity to positive infinity. So we found given the graph that this is a function and both its domain and range are from negative infinity to positive infinity, which corresponds with answer choice D. Thanks everyone for watching. I hope this video helped see you in the next one.

Determine whether each relation defines a function, and give the domain and range. See Examples 1–4.

Key Concepts

Definition of a Function

Domain of a Function

Range of a Function

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