Answer
Yes, r(t) is one-to-one.
Work Step by Step
The function $r(t)=t^6-3$, $0\leq t\leq 5$ is one-to-one because it passes the horizontal line test on the given interval (the graph is an even function, but the interval only spans one side of it). We can show this algebraically:
We start with the assumption that:
$t_1\ne t_2$
Then:
${t_1}^6\ne {t_2}^6$
(Since raising a unique positive number to the 6th power results in a unique value.)
${t_1}^6-3\ne {t_2}^6-3$
So the function would not have the same $y$ value for two different $t$ values (one-to-one).
