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

Question

Determine whether each relation defines a function, and give the domain and range.

Graph of an ellipse centered at the origin with x-intercepts at ±4 and y-intercepts at ±3 on a grid.

Accepted Answer

Mhm Hey, everyone in this problem, we're asked to identify if the given graph represents a function and also define the domain and range. Now, the graph are given shows something that looks like an ellipse and it has a maximum Y value at eight, a minimum Y value at negative eight, a maximum X value of five and a minimum X value of negative five. So and it's, it's an ellipse centered at zero. We're given four answer choices. Option A, it is a function. The domain is from negative 5 to 5 inclusive and the range is from negative 8 to 8 inclusive. Option B it is a function. The domain is from negative 8 to 8 inclusive and the range is from negative 5 to 5 inclusive. Option C it's not a function. The domain is negative 5 to 5 inclusive and the range is negative 8 to 8 inclusive. And option D it's not a function. The domain is negative 8 to 8 inclusive and the range is negative 5 to 5 inclusive. So when we're given a graph and we want to find out whether or not it represents a function, what we wanna do is we wanna perform our vertical line test. And what our vertical line test does is we go to our graph and we draw a vertical line anywhere we want. OK. And for any, that's almost vertical, for any vertical line, we draw, it should only hit the graph in one point. While in our graph, if we draw a vertical line, it actually hits this curve in two different points and one at the top one at the bottom. And so this graph does not pass the vertical line test. I think it's a red X for the vertical line test. What this means is that for every X value, we have, we have more than one corresponding Y value. So this is not a function. OK. So it does not pass the vertical wine test which tells us that it is not a function. All right. So we can already narrow down our answer choices to two. We've found that this is not a function. And so we can eliminate option A and option B where we find that it is a function. OK? So it's not a function, but let's look at domain and range. Now the domain is gonna be all of the possible X values. OK. So if we're looking at the domain, we can see that the X values range from negative five up to five. OK? We don't have any X values further left or further right than those values. And so our domain is gonna be from negative 5 to 5. Ok. And we need to figure out whether to use closed brackets. Ok, round brackets or square brackets. Now, in this case, we're using square brackets, square brackets indicate that we include the end points. Why do we want to include the end points? Well, we can have an X value of negative five and we can have an X value of positive five, just nothing bigger. So we want to include those end points. We use square brackets. Similarly, for the range, the range is gonna be all of the possible Y values. OK? We have a minimum Y value at negative eight and nothing below it. We have a maximum Y value at positive eight and nothing above it. So our range is gonna be from negative eight to positive eight. And we're gonna use square brackets again because we can have those values of negative at eight and eight, just nothing further. So we include our end points. And so using the vertical line test, we found that the graph does not represent a function. We found that the domain is from negative 5 to 5, including the end points. And the range is from negative 8 to 8 including the end points. This corresponds with answer choice. C 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.

Key Concepts

Definition of a Function

Domain of a Relation

Range of a Relation

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