Question

How do I show f(x,y)= x+y is surjective?

Answer 1

so hard for me

Answer 2

Surjective is (onto). The domain is \(\mathbb{R}^2\), because \(x,y\in\mathbb{R}\) and the codomain is \(\mathbb{R}\).

Answer 3

So, we need to show that this two-variable function maps all real numbers.

Answer 4

I don't want to post any official proves, but (one possible way) ...

You know that this function certainly satisfies \(x=y\) for any value of \(x\).
(Because the domain is all real numbers.)
Thus, you know that for any real number \(x\), you have \(z=2x\), or \(\frac{z}{2}=x\)
And since this is for all real numbers \(x\), therefore all real numbers \(z\) are included as well.

Answer 5

(This is not a rigorous proof of course, but just a "thinking" ...)

Answer 6

Thank you for explaining everything so clearly!