Question

Give examples of two surjective functions f₁ and f₂ from Z to Z such that f₁ + f₂ is not surjective.

Answer 1

We know that f₁: R → R, given by f₁(x) = x, and f₂(x) = -x are surjective functions.
Proving f₁ is surjective :
Let y be an element in the co-domain (R), such that f₁(x) = y.
f₁(x) = y
$\Rightarrow$x = y, which is in R.
So, for every element in the co-domain, there exists some pre-image in the domain.1(x)=f1(y)x=y
So, f₁is surjective .

Proving f₂ is surjective :Let f2(x)=f2(y)−x=−yx=y
Let y be an element in the co domain (R) such that f₂(x) = y.
f₂(x) = y
$\Rightarrow$x = y, which is in R.
So, for every element in the co-domain, there exists some pre-image in the domain.1(x)=f1(y)x=y
So, f₂ is surjective .

Proving (f₁ + f₂) is not surjective :
Given:
(f₁ + f₂) (x) = f₁ (x) + f₂ (x)= x + (-x) =0
So, for every real number x, (f₁ + f₂) (x)=0
So, the image of every number in the domain is same as 0.
$\Rightarrow$Range = {0}
Co-domain = R
So, both are not same.
So, f₁ + f₂ is not surjective.