Find functions with given domain and range

[xsw001]

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 continuous 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?

 

[quantumdude]

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 $f(x)=tan(ax+b)$ and figure out what $a$ and $b$ should be.

2) Find a continuous 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.

 

[xsw001]

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.

 

[quantumdude]

Why not use sine or cosine?

 

[xsw001]

Oh, yeah, thanks!