Just a question about vertical and horizontal line tests

In summary, the conversation discusses the use of the vertical and horizontal line test to determine if a graph represents an injective or surjective function. The tests are not always accurate and it is sometimes better to show it algebraically. The example of f(x)=x^2+4x-5 is used to demonstrate how to determine if a function is injective or surjective. The conversation also highlights the importance of understanding the domain and range of a function in determining its attributes.
  • #1
lionely
576
2
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
lionely said:
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
Hm... if it cuts it twice like 1 value of y can be mapped to x so it's not 1 to 1?
 
  • #4
lionely said:
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
When do the line tests fail?
 
  • #6
lionely said:
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
Like I once heard my teacher say when it fails umm show it's one to one or w/e algebraically.
 
  • #8
lionely said:
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
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
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
lionely said:
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
Determine whether or not g is onto f(x)=x^2+4x-5, x>-2
 
  • #13
Oh and thank u 5hassay for that explanation
 
  • #14
lionely said:
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
0,-5. It is not one to one
 
  • #16
lionely said:
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
The vertex is -2,9
 
  • #18
lionely said:
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
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
lionely said:
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
Umm is it that it's only surjective for a certain range
 
  • #22
lionely said:
Umm is it that it's only surjective for a certain range

I would agree with that. What range?
 
  • #23
Umm x<-2?
 
  • #24
lionely said:
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
The domain is x>-2. Range f(x) >/ -9? Since range doesn't equal domain its not surjective?
 
  • #26
lionely said:
The domain is x>-2. Range f(x) >/ -9? Since range doesn't equal domain its not surjective?

Ok, if the domain is x>-2. That means f(x)>(-9). I think the implied range is probably (-infinity,infinity) if they don't tell you anything else. That would mean -10 is not a value of f(x). What does that tell you about surjectivity? And is it injective?
 
  • #27
I think it's injective when x>-2 so yes it's injective and not surjective but it would be surjective if the range. Was the same as the domain
 
  • #28
lionely said:
I think it's injective when x>-2 so yes it's injective and not surjective but it would be surjective if the range. Was the same as the domain

Yes, it's injective if the domain is (-2,infinity). If you were given the range is (-9,infinity) then it would also be surjective. If you were given that the range is (-2,infinity) then you have a serious problem. f isn't even a function because f(-2) isn't in the range. Try to get off this 'domain=range' means surjective thing.
 
  • #29
Oh I was saying domain =range because I saw it in the textbook =/ I am going to try and practice this function thing til I get it
 

1. What is the purpose of the vertical and horizontal line tests?

The vertical and horizontal line tests are used to determine if a given graph represents a function. They help to identify any vertical or horizontal lines that may intersect the graph at more than one point, which would indicate that the graph is not a function.

2. How do you perform the vertical and horizontal line tests?

To perform the vertical line test, you simply draw a vertical line through the graph. If the vertical line intersects the graph at more than one point, then the graph is not a function. To perform the horizontal line test, you draw a horizontal line through the graph. If the horizontal line intersects the graph at more than one point, then the graph is not a function.

3. Why are the vertical and horizontal line tests important in mathematics?

The vertical and horizontal line tests are important because they help to determine if a graph represents a function. Functions are essential in mathematics as they are used to model real-world situations and solve problems in various fields such as physics, economics, and engineering.

4. Are there any exceptions to the vertical and horizontal line tests?

Yes, there are some exceptions to the vertical and horizontal line tests. Some graphs, such as circles, do not pass the vertical line test but are still considered functions. This is because each input on the graph has only one corresponding output.

5. Can the vertical and horizontal line tests be used to determine the domain and range of a function?

Yes, the vertical and horizontal line tests can help determine the domain and range of a function. The domain of a function is the set of all possible input values, which can be identified by using the vertical line test. The range of a function is the set of all possible output values, which can be identified by using the horizontal line test.

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