Types of functions (one-to-one)

glenn101 · 2008-12-23 15:30:37

Ok, my questions revolve around asking whether a function is one-to-one.

From my understanding one method of testing whether a function is one-to-one is by using the horizontal line test which implies that a function is one-to-one if it crosses the graph at only one point then the function is known as a one-to-one.

However, why isn't the graph with relation {(x,y): y^2=x+2, x(greater than or equal to) -2} a one-to-one?
This graph looks like a sideways parabola and the horizontal line crosses the graph at only one point! however I checked the answer and it isn't a one-to-one?!?! what is wrong?
If a graph is not a function, does that mean it isn't one-to-one?

Does the horizontal line test allow every linear graph to be one-to-one? is every linear graph a one-to-one function?

on another note, does R(plus) union{0} mean the values of R including 0?

and here is my final question.
a) Draw the graph of g:R->R, g(x)=x^2+2
I've drawn that, it is a parabola which has been translated 2 units vertically.
b)By restricting the domain of g, form two one-to-one functions that have the same rule as g.
ok I get that g1(x)=x^2+2, x(greater than or equal to) 0
but it says g2(x)=x^2+2, x<0. Shouldnt that be (less than or equal to) 0

I'm very confused by all of this, any help would be much appreciated.
Thanks,
Glenn.

Ricky · 2008-12-23 18:19:47

If a graph is not a function, does that mean it isn't one-to-one?

Typically, 1-1 only refers to functions. If you are studying relations, you may be able to call a relation that is not a function (in other words, it isn't well-defined) 1-1, but I have personally never heard this. You aren't studying relations, so it is most likely that 1-1 will only refer to functions.

Does the horizontal line test allow every linear graph to be one-to-one? is every linear graph a one-to-one function?

y = 0

on another note, does R(plus) union{0} mean the values of R including 0?

If the "(plus)" is there, then it means the positive values and 0. Another word for this is the "nonnegative" values. Remember, as a mathematician you are allowed to stick "non" on to the front of any word, even if it is noncommon to do so.

b)By restricting the domain of g, form two one-to-one functions that have the same rule as g.
ok I get that g1(x)=x^2+2, x(greater than or equal to) 0
but it says g2(x)=x^2+2, x<0. Shouldnt that be (less than or equal to) 0

In fact, you may include or not include 0 and in either case you will wind up with two 1-1 functions. Or you could restrict the domain to [2, 5] or [-6, -2.835828572572852]. There are infinitely many solutions. The solution you have gives the largest two nonoverlapping (there's that "non" again!) solutions.

glenn101 · 2008-12-24 12:34:14

Ricky- We are studying relations as well but I think that one-to-one only applies to functions in our case
.
We are using the text book "Mathematical Methods CAS 3&4 by Cambridge, chapter 1 this is from".

Thanks for the help Ricky, you seem to have exceptional mathematical knowledge

Macy · 2008-12-24 13:01:16

To answer your question about function and relation
1. We must have relation first before we can classify two sets of (x,y) pairs is function or not
2. From relation to become function it must fall into the 1:1 or many :1 relationship
3. Linear is always function as it names because it always have 1:1 relationship
4. We can use the horizontal or vertical line to test whether the relation is a function depending on what you define, however most of the books only talk about horizontal line.
5. Only when we test whether the function has the inverse function or not then we use the horizontal line ONLY.

John E. Franklin · 2008-12-30 09:11:09

Might also an ever increasing or always decreasing function be needed for 1 to 1 in both directions, those being, x to y and also y to x ??

Ricky · 2008-12-30 09:42:10

John E. Franklin wrote:

Might also an ever increasing or always decreasing function be needed for 1 to 1 in both directions, those being, x to y and also y to x ??

Any function that is 1-1, it's inverse defined on the image of f is 1-1 as well. Turns out the inverse being 1-1 is equivalent to the original function being well-defined.

Functions need not be always increasing or decreaing to be 1-1. An easy example of this is f(x) = x except at -1 and 1 where f(x) = -x.

Math Is Fun Forum

#1 2008-12-23 15:30:37

Types of functions (one-to-one)

#2 2008-12-23 18:19:47

Re: Types of functions (one-to-one)

#3 2008-12-24 12:34:14

Re: Types of functions (one-to-one)

#4 2008-12-24 13:01:16

Re: Types of functions (one-to-one)

#5 2008-12-30 09:11:09

Re: Types of functions (one-to-one)

#6 2008-12-30 09:42:10

Re: Types of functions (one-to-one)

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