Answer
Yes.
Work Step by Step
Test to show that $f(a)=f(b)$ implies $a=b$.
$3a^3-6=3b^3-6$
Add 6 to each side.
$3a^3=3b^3$
Divide each side by 3.
$a^3=b^3$
Take cube root of each side.
$a=b$; Yes
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