College Algebra 7th Edition

Published by Brooks Cole

Chapter 2, Functions - Section 2.8 - One-to-One Functions and their Inverses - 2.8 Exercises - Page 262: 21

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).

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