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

Question

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

Graph showing a point at (-2, -4) with arrows indicating a relation.

Accepted Answer

Hello everybody. I have everything right today. Today we're gonna be looking at this math question that's asking us to identify if the given graph represents a function and also find alumin and range. Now, the graph that they give us is a parabola on its side. So it's opening sideways to the uh towards the left hand side. And this is in quadrant three of a graph with an origin of negative two comma negative four. Now the provided say A says that it is not a function with a domain of open parentheses, negative infinity comma negative two with a bracket, closing bracket and a range of negative negative infinity to positive infinity B says not a function with the domain of open parentheses, negative affinity comma negative four and a closing bracket. And uh the same range as A C says function with the domain and range, the same as letter A and D says function with the domain and range the same as B. Now, the first thing we gotta think of is what is a function, right? They're asking us to identify if this is a function. So what exactly is a function? How can we know if this is a function, well, a basic definition of what a function is, is just basically whatever a function will have one output value for each input value that you have. So let's think about it. Usually when you're talking about function, um it's usually written as F of X, right, the function of X is equal to something, right? So that X would be your input value in this case. So for every X that you input into this function, you're gonna have one result, one answer. So let's apply that logic to our graph. Let's say we try X of negative four, right. So if we put, if we are the X are at the X of negative four, and we see where our graph intersects the X of negative four, we'll see that inter it intersects it at around negative three, that's one intersection. But then we also see that it intersects at around a Y of negative five. So that means there's two output values for this one input. Therefore, this is our first indication that this is not a function another way to test this. And which is more common is called the vertical line test, right? So basically the vertical line test is exactly what it says. You draw a vertical line up and down your graph and see how many times you intersect the graph. If you intersect your graph more than one time, then this is not a function because uh once again, that means you're, you're having two outputs for one X input. So let's do exactly that. Let's just draw some vertical lines down our graph. And I'll draw one in negative four, X negative six and X negative eight. And we see that for each of those vertical lines, we're hitting our graph twice because this is a sideways parabola. So we hit the graph twice once in the, on the upper side of the Parabola and one on the lower side of the parabola. So that confirms that this is not a function. Therefore, we can eliminate right away, answer choice C and answer choice D. And now we're left with finding the range of the, and the domain of the function while the domain has to do with X values. So the domain is any possible X values that the function will have. Well, we see that this is a parameter that goes, that starts in a quadrant three and is opening to the left hand side and just continues infinite infinitely. Therefore, we will have negative infinity. That'll be our first domain. It'll be from, from negative infinity comma. And then we see that the origin of the parabola is that X negative two with a closed origin. Therefore, it will be from negative two to, from negative infinity to negative two. And it'll be a bracket C closing, not a parentheses closing because we do start a closed point of negative two So it includes negative two. If we had an open parenthesis of the negative two, that means we're not including negative two. That's why we ride a bracket. So that would eliminate answer choice B and we're left with a choice A. So let's just confirm the range just to make sure that it's correct. Now, the range, unlike the domain has to do with the Y values, any possible Y values of the function And of the graph. Well, we see that the problem is on its side once again and it's opening up and opening down with arrows on each side. So imagining this goes on infinitely, those two sides of the parabola will go infinitely up and infinitely down. Therefore, we will have a range, a range from negative infinity to positive infinity. Because once again, this, this, this, this has to do with the Y values, we can open up infinitely upwards and infinitely downwards. So that confirms that answer choice. A is the answer to this question. So I hope that was helpful. And until next time.

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

Key Concepts

Function Definition

Domain and Range

Graph Interpretation

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