Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 2 - Section 2.8 - One-to-One Functions and Their Inverses - 2.8 Exercises - Page 227: 66

Answer

$f^{-1}(x)=\sqrt[5]{\sqrt[7]{x}+6}$

Work Step by Step

$f(x)=(x^{5}-6)^{7}$ Rewrite this expression as $y=(x^{5}-6)^{7}$ and solve for $x$: $y=(x^{5}-6)^{7}$ Take the seventh root of both sides: $\sqrt[7]{y}=\sqrt[7]{(x^{5}-6)^{7}}$ $\sqrt[7]{y}=x^{5}-6$ Take $-6$ to the left side: $\sqrt[7]{y}+6=x^{5}$ $x^{5}=\sqrt[7]{y}+6$ Take the fifth root of both sides: $\sqrt[5]{x^{5}}=\sqrt[5]{\sqrt[7]{y}+6}$ $x=\sqrt[5]{\sqrt[7]{y}+6}$ Interchange $x$ and $y$: $y=\sqrt[5]{\sqrt[7]{x}+6}$ The inverse of the original function is $f^{-1}(x)=\sqrt[5]{\sqrt[7]{x}+6}$
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