Sorry, I know that it has to be a very simple problem, but I'm frustrated because of it.
Let $f,g:\mathbb{N}^3→\mathbb{N}f$:
$f(x,y,z)=3^x⋅5^y⋅7^z$ and $g(x,y,z)=3^x+5^y+7^z$ Prove that:
$1.f$ is injective but $g$ isn't.
$2.$ Prove that they aren't bijective.
EDIT: those were power, not multiplications.
EDIT 2: sorry, I'm the OP, but I don't know how to edit the question as the original poster, so it appeared as Community :(
Fill in details:
$$g(2,2,2)=g(1,4,1)$$
and also
$$f(3,5,1)=f(5,3,1)$$
Thus, neither $\;f\;$ nor $\;g\;$ are injective.
Also observe that
$$f(x,y,z)\ge 21\;,\;\;g(x,y,z)\ge 15\implies\text{neither function is surjective}$$