Just a question about vertical and horizontal line tests

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  • #1
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There's no real question per say. I've been looking in books and on the internet i can't find anything.

I don't know when to use the vertical and horizontal line test to test for injective functions and surjective.

Let's say if I was to sketch the graph of x2+4x - 5

I would get a parabola, it's a function cause a vertical line cuts it once. Now a horizontal line cuts it twice... so what does that mean? It's surjective?
 

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  • #2
Dick
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There's no real question per say. I've been looking in books and on the internet i can't find anything.

I don't know when to use the vertical and horizontal line test to test for injective functions and surjective.

Let's say if I was to sketch the graph of x2+4x - 5

I would get a parabola, it's a function cause a vertical line cuts it once. Now a horizontal line cuts it twice... so what does that mean? It's surjective?

If a horizontal line cuts it twice that means it's not injective. But not all horizontal lines cut it twice. Some don't cut it at all. What does that mean? Think back to what 'injective' and 'surjective' really mean. That will explain why the line test works.
 
  • #3
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Hm... if it cuts it twice like 1 value of y can be mapped to x so it's not 1 to 1?
 
  • #4
Dick
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Hm... if it cuts it twice like 1 value of y can be mapped to x so it's not 1 to 1?

If you mean if it cuts twice then two values of x correspond to one value of y, then yes, not injective.
 
  • #5
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When do the line tests fail?
 
  • #6
Dick
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When do the line tests fail?

Why would you ask that? What makes you think they do? What exactly are the 'line tests' you've been taught?
 
  • #7
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Like I once heard my teacher say when it fails umm show it's one to one or w/e algebraically.
 
  • #8
Dick
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Like I once heard my teacher say when it fails umm show it's one to one or w/e algebraically.

That's kind of vague. Like, sort of.
 
  • #9
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The horizontal and vertical line test is an intuitive way to see if a graph belongs to an injective function. To be clear, if the graph belongs to a function, and a horizontal line drawn anywhere on the graph means that the line will only intersect the graph at most once, then the function is injective, that is, the function evaluated at any two points means that the evaluation will be different for these two points. Continuing, a graph belongs to a function if a vertical line can be drawn anywhere on the graph and it will happen that the line intersections the graph at maximum one time. The idea here is that a graph is only a function if an evaluation at one point means you are only going to get one evaluation, and not, say, two.

This should make everything clear enough so that you can do that example yourself. If not, say so, :).
 
  • #10
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Dick I mean I heard my teacher say the tests aren't always accurate so it's better to show it algebraically. Also I I believe I have a problem showing a function is surjective algebraically
 
  • #11
Dick
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Dick I mean I heard my teacher say the tests aren't always accurate so it's better to show it algebraically. Also I I believe I have a problem showing a function is surjective algebraically

If your teacher means that you might not be able to draw graphs accurately enough to be sure, then that makes sense. Which function are you having problems with?
 
  • #12
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Determine whether or not g is onto f(x)=x^2+4x-5, x>-2
 
  • #13
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Oh and thank u 5hassay for that explanation
 
  • #14
Dick
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Determine whether or not g is onto f(x)=x^2+4x-5, x>-2

Did you look at the graph? What did you decide from that? You said you got a parabola. Where was the vertex?
 
Last edited:
  • #15
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0,-5. It is not one to one
 
  • #16
Dick
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0,-5. It is not one to one

(0,-5) is not the vertex, but it IS not 1-1. Give me a better reason. If you complete the square you can show f(x)=(x+2)^2-9. Does that help?
 
  • #17
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The vertex is -2,9
 
  • #18
Dick
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The vertex is -2,9

No, it's not. It's (-2,-9). But I'm not really even interested in that. I am interested in what you think about surjectivity.
 
  • #19
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Oh I meant to type-9 I'm on a mobile device but um it's surjective since mOre two things in x map to one y.
 
  • #20
Dick
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Oh I meant to type-9 I'm on a mobile device but um it's surjective since mOre two things in x map to one y.

Oh, come on. Surjective doesn't have anything to do with two things mapping onto one. Look up surjective on the mobile device.
 
  • #21
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Umm is it that it's only surjective for a certain range
 
  • #22
Dick
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Umm is it that it's only surjective for a certain range

I would agree with that. What range?
 
  • #23
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Umm x<-2?
 
  • #24
Dick
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Umm x<-2?

x<-2 is a domain. What range? This is getting silly. You know what the graph looks like. You can restrict the domain and range to make it surjective or injective or both or neither. What ARE the domain and range?
 
  • #25
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The domain is x>-2. Range f(x) >/ -9? Since range doesn't equal domain its not surjective?
 

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