Function counter-image...

Question

Given the function:

$g \circ f: |n^2 - 1| + 1$

which is a composed function of

$f: n ∈ Z → n^2 - 1 ∈ Z$

$g: m ∈ Z → |m| + 1 ∈ N$

Is this correct:

$g \circ f^{-1} (\{4\}) = ±2$ ?

Answer 1

If $$ |n^2-1|+1 = 4 $$

Then $$ |n^2-1| = 3 $$

But $n^2 - 1 \ge -1$, therefore $$n^2-1 = 3$$

or $$ n^2 = 4 \implies n = \pm 2$$