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Find functions with given domain and range

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1) Find a continuous function f: (0,1)->R with f[(0,1)]=R
I couldn't think of any function except tangent, but its domain is NOT (0,1) though? Any suggestions?

2) Find a continous function f: (0,1)->R with f[(0,1)]=[0,1]
I couldn't think of any function that I know. Any suggestions?

3) Find a continuous function f: R->R this is strictly increasing and f(R)=(-1,1)
The graph is somewhat look like f(x)=x^(1/3), but not exactly though since the domain doesn't fall into (-1,1). Any suggestions?
 

Tom Mattson

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1) Find a continuous function f: (0,1)->R with f[(0,1)]=R
I couldn't think of any function except tangent, but its domain is NOT (0,1) though? Any suggestions?
Not a bad idea. Modify it to [itex]f(x)=tan(ax+b)[/itex] and figure out what [itex]a[/itex] and [itex]b[/itex] should be.

2) Find a continous function f: (0,1)->R with f[(0,1)]=[0,1]
I couldn't think of any function that I know. Any suggestions?
You're being asked to find a function whose range is [0,1]. Surely you can immediately come up with 2 examples.

3) Find a continuous function f: R->R this is strictly increasing and f(R)=(-1,1)
The graph is somewhat look like f(x)=x^(1/3), but not exactly though since the domain doesn't fall into (-1,1). Any suggestions?
I'm thinking arctangent here.
 
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Okay, I got the first and third one, but can you provide me some oscillated function for the 2nd one though? I know it has to do with 1/[x(x-1)], but the graph is not exactly in between [0,1] though.
 

Tom Mattson

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Why not use sine or cosine?
 
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Oh, yeah, thanks!
 

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