f: [0, ∞) → R

L

lc212

New member
Joined
Apr 17, 2022
Messages
2
Hello, I have a question and I hope somebody can help me answer it;

Can function f: [0, ∞) → R be surjective?
 
Hello LC. Have you learned the meaning of surjective, yet? Is this part of a class assignment? Please share your thoughts about the exercise. Thank you.

Posting Guidelines (Summary)

Welcome to our tutoring boards! :) This page summarizes the main points from our posting guidelines. As our name implies, we provide math help (primarily to students with homework). We do not generally post immediate answers or step-by-step solutions. We don't do your homework. We prefer to...
www.freemathhelp.com
 
lc212 said:
Hello, I have a question and I hope somebody can help me answer it;

Can function f: [0, ∞) → R be surjective?
Click to expand...
Have you tried making (or at least imagining) such a function? Sketch a graph of what it might look like. If you can't, tell us what is preventing you, and we can discuss that.
 
Have a look at this graph.
[imath]y=k\in\Re[/imath] is a horizontal line along which every point looks like [imath](x,k)[/imath]
Using the graph in the link, would every such line intersect that graph?
What does that have to do with your question?
 
Last edited by a moderator:
blamocur said:
Does [imath]f[/imath] have to be continuous?
Click to expand...
Yes.

Dr.Peterson said:
Have you tried making (or at least imagining) such a function? Sketch a graph of what it might look like. If you can't, tell us what is preventing you, and we can discuss that.
Click to expand...
Yes, I tried and I can't imagine it.

mmm4444bot said:
Hello LC. Have you learned the meaning of surjective, yet? Is this part of a class assignment? Please share your thoughts about the exercise. Thank you.

Posting Guidelines (Summary)

Welcome to our tutoring boards! :) This page summarizes the main points from our posting guidelines. As our name implies, we provide math help (primarily to students with homework). We do not generally post immediate answers or step-by-step solutions. We don't do your homework. We prefer to...
www.freemathhelp.com
Click to expand...
Yes, we have learned all about functions, already in high school and again now. It is actually a question from the theoretical part of the exam at my university (I study pharmacy). We have to decide if the statement is correct and prove our decision, and I cannot imagine such function and do not know how to solve it, so I decided to ask. English is also not my first language, sorry for any mistakes.
 
lc212 said:
Hello, I have a question and I hope somebody can help me answer it;

Can function f: [0, ∞) → R be surjective?
Click to expand...
Please tell us - what does it mean for a function to be surjective?
 
lc212 said:
Yes, I tried and I can't imagine it.
Click to expand...
Sometimes in math you have to develop a good imagination, and not let it be limited to the familiar. We can get so used to polynomials that we forget there are other kinds of function!

Think about it: assuming (though it wasn't stated) that the function has to be continuous, it can't be @pka's log (which isn't defined at 0); that suggests it has to grow both upward and downward as x increases, perhaps starting at the origin. So some sort of oscillating function seems necessary.

You don't need to give the equation of an actual function, but @blamocur's example exactly fits these conditions.
 
@lc212 had you given the complete question(i.e. continuous) I would modified the answer.
[imath]f(x)=\log(x)\text{, if}\; x\le 1\\\;\;\;\;\;\;\;\:=x-1\;\text{, if }\; x> 1[/imath]
That is somewhat easier to see the horizonal test for surjectivity.
 
You must log in or register to reply here.
Top