Is the set of functions: {[math]f® = R |\frac{df}{dx} + 2f = 1[/math]} a vector space? I said no because it doesn't seem to have a zero vector, but I'm doubtful of my answer. Can someone help me prove its vector space validity (or lack thereof)?

I'm not quite sure on my interpretation of your question. Are you asking for all functions f that have a fixed point at R, or the fixed points of the function f, or what?

No, I mean f of a real number is another real number. But I have no idea how to make the funky looking R with Latex

Are you trying to say the following?

 

[math]\left\{f:\mathbb{R}\rightarrow\mathbb{R}\mid\frac{df}{dx}+2f=1\right\}[/math]

 

If so then you are correct. [imath]f\equiv0[/imath] doesn't satisfy the condition specified in the set definition.

 

No, I mean f of a real number is another real number. But I have no idea how to make the funky looking R with Latex

 

Type the following, without the spaces:

 

[ math ]\mathbb{R}[ \math ]

Yeah, that's what I meant, thanks!