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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 61 (page 1005)

In the following exercise, find the inverse of each function.
$f (x) = \sqrt[5]{- 3 x + 5}$

Short Answer

Expert verified

The inverse of the given function is $f^{- 1} (x) = \frac{5 - x^{5}}{3}$

Step by step solution

01

Step 1. Given information

The given function is $\sqrt[5]{3 x - 5}$

02

Step 2. Interchange the variable x and y

Substituting $f (x) = y$ , we get:

$f (x) = \sqrt[5]{- 3 x + 5}$

Interchanging the variable $x$ and $y$ , we get:

$x = \sqrt[5]{- 3 y + 5}$

03

Step 3. solve for y

Solving for $y$ , we get:

$x = \sqrt[5]{- 3 y + 5} x^{5} = - 3 y + 5 - 3 y = x^{5} - 5 y = \frac{5 - x^{5}}{3}$

Substitute $y = f^{- 1} (x)$ , then,

role="math" localid="1644935105081" $f^{- 1} (x) = \frac{5 - x^{5}}{3}$

04

Step 4. Verify that the functions are inverses.

Substituting $f (x)$ in $f^{- 1} (f (x)) = x$ , we get:

$f^{- 1} (f (x)) = x f^{- 1} (\sqrt[5]{- 3 x + 5}) = x \frac{5 - {(\sqrt[5]{- 3 x + 5})}^{5}}{3} = x \frac{5 - (- 3 x + 5)}{3} = x \frac{3 x}{3} = x x = x$

This is true.

Substituting $f^{- 1} (x)$ in $f (f^{- 1} (x)) = x$

$f (f^{- 1} (x)) = x f (\frac{5 - x^{5}}{3}) = x \sqrt[5]{- 3 (\frac{5 - x^{5}}{3}) + 5} = x \sqrt[5]{- 5 + x^{5} + 5} = x \sqrt[5]{x^{5}} = x x = x$

This is also true.

Thus, the obtained inverse is correct.

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Most popular questions from this chapter

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