Question

f(x)=2x+|cosx|
Explain the nature of f(x),i.e onto/into & one-one/many one.

Answer 1

cos pi and cos 0 both give the same value ..so It shouldn't really be one-one
but 2x+|cosx| are both forever increasing too..so it should be one-one
I'm confused about this.

Answer 2

@agent0smith and @terenzreignz

Answer 3

i think the 1st statement is wrong since 2x will have different values so it should be one one?

Answer 4

if the question was 2+|cosx| then it should've been many one if I'm right? since cos 0 and cos pi will point to the same value

Answer 5

http://math.stackexchange.com/questions/101975/how-to-determine-if-a-function-is-one-to-one

This gives an algebraic method for proving it's one to one.

Answer 6

If every element in the range corresponds to exactly one element in the domain, then it is one-to-one, obviously. This function is because of the 2x term, which is linear, with a slope of 2. This function is a curve that behaves like a straight line with a slope.

Answer 7

f'(x)=0

Answer 8

f(x) = 1

Answer 9

What is into?

Answer 10

It should be a curve thats a straight like in the 3rd quad but with a slope in the 1st quad. |cosx| wont interfere with 2x's graph in the third quad right?

Answer 11

Graph:

https://www.google.com/search?q=2x%2Babs(cos(x))&aq=f&oq=2x%2Babs(cos(x))&aqs=chrome.0.57j0j62l3.1313j0&sourceid=chrome&ie=UTF-8

Answer 12

The google graph clearly indicates this to be both one-to-one and onto.

Answer 13

I am just a lil confused as to why |cosx| would interfere with 2x's graph in the third quad. Or is my approach if imagining graphs like this is wrong?

Answer 14

good question^

Answer 15

click the link given by @agent0smith and you can see how the function looks.

Any function that is onto is into, but not the other way around. Onto states that every element in the range (codomain) matches with a unique element in the domain.

An INTO function is one in which all the elements of the domain match to at least one element in the codomain.

ONE-TO-ONE functions are functions in which every element in the domain exactly matches a unique element in the codomain.

Answer 16

But don't that mean that all functions are into?
I mean, if an "element of its domain" has no match in the codomain, then that is not an element of its domain...

Answer 17

That depends on the range. The reason for the question is that if the function is not ONTO then it is only INTO if at least one element of the domain matches to at least one element of the range.

Answer 18

You won't ever come across a function that is not at least INTO.

Answer 19

Okay, that clears things up... Danke, Herr Euler :D

Answer 20

unless it is specified as such. But such functions are useless.

Answer 21

But then again, if all functions are into, why stress it? :D

Answer 22

Has anyone heard of the method to find whether a function is onto or into by putting
\[f'(x)=0 ?\]

Answer 23

Doesn't that only give you info on local extrema?

Answer 24

But I suppose...

Answer 25

Here is an example of a function that is not INTO:

f(x) = x, where the domain is the empty set.

It is a legitimate enough function, but not of much use.

Answer 26

Ich verstehe, Herr Euler :D
(I understand)

Answer 27

I have not heard of using the derivative to determine onto, but it is an idea worth checking out. The derivative is a very useful tool. Once again, the domain and range are important in determining these properties.

Answer 28

Can anyone tell If I'm right ?
The range of f(x) tells whether the function is onto or into
and The range of f'(x) tells whether the function is onto or into
?

Answer 29

i heard if \[\LARGE f'(x)\] is positive then the functions is forever increasing so..

Answer 30

im confused,anyway the question is solved,thanks!

Answer 31

I dont think f'x can help with onto-into.. it can help with one-to-one or many to one though.