Answer
Yes.
Work Step by Step
Test to show that $f(a)=f(b)$ implies $a=b$.
$2a^3-1=2b^3-1$
Add 1 to each side.
$2a^3=2b^3$
Divide each side by 2.
$a^3=b^3$
Take cube root of each side.
$a=b$; Yes
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