Functions

**Almaniac** · January 6th 2011, 07:37 AM

How we prove in a function f that f(R)=R?
By proving that is '1-1'?

**emakarov** · January 6th 2011, 07:50 AM

If $f:\mathbb{R}\to\mathbb{R}$ , i.e., it is declared that the domain and the codomain of $f$ are $\mathbb{R}$ , then the claim that $f(\mathbb{R})=\mathbb{R}$ means that $f$ is surjective. On the other hand, the fact that $f$ is 1-1 means that it is injective. These are different concepts; none of them implies the other. For example, for $f(x)=x^3-x$ we have $f(\mathbb{R})=\mathbb{R}$ , but $f$ is not 1-1.

**Almaniac** · January 6th 2011, 07:53 AM

thank you man

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