How does surjectivity and injectivity apply to multi-input (multivariable) function?
Let say we have a function from R x R to R:
f(x,y) = x + y + 6
If we assume that f(x,y) = 9, so:
9 = x + y + 6
And thus:
3 = x + y
So does it mean that the function is not injective for pairs x,y (for example) f(1.5, 1.5) and f(0, 3) as both are giving the same output equal 3? Or is it not working that way?
And in terms of surjectivity, if x, y ∈ R and f(x,y) = z , function will be surjective if the output will simply be z ∈ R?
Best regards
Best Answer
The function is surjective if for every element of the codomain, say $z$, there exists an element of the domain whose image is $z$. In this case, $f$ is surjective since for each $z \in \mathbb R$ we have $f(z,-6) = z$. The injectivity part is fine!