a question about proofing surjective

- let f :R{1,-1}-R and f(x)=x/x
^{2}-1. - I am trying to show that the function is surjective,but i have the following questions:
- 1）why we need to divide the case into 2 parts (1):y:R\{0} and (2)y=0 before proving it is surjective.
- 2）and for part(1),why we need to show x:R\{1,-1}. I.e. x isn't equal to +1/-1
- 3）BTW,for the definition of surjective,i don't know the pricinple behind f(x)=y (or why we need to show f(x)=y) (for all y£Y,the exists x£X s.t. y=f(x)

It would be great if you could help me

Re: a question about proofing surjective

Quote:

Originally Posted by

**sethplau** - let f :R{1,-1}-R
- F(x)=x/x
^{2}-1. - I am trying to show that the function is surjective,but i have the following questions:
- 1）why we need to divide the case into 2 parts (1):y:R\{0} and (2)y=0
- 2）and for part(1),why we need to show x:R\{1,-1}. I.e. x isn't equal to +1/-1
- 3）and for the definition of surjective,i don't know the pricinple behind f(x)=y (or why we need to show f(x)=y) (for all y£Y,the exists x£X s.t. y=f(x)

I am not really sure what you want done.

Consider $\displaystyle f: (-1,1)\to\mathbb{R}$ defined be $\displaystyle f(x)=\frac{x}{x^2-1}$.

That is a continuous onto function.

Are you asking to have the onto part proven?

Re: a question about proofing surjective

I want to know how to prove the function is surjective.

Though I've seen the detailed solution which show that the proof should be divied into 2parts

， i don't really know why it should be proved through 2parts~