Chapter 10: Q. 10.18 (page 1001)

Find the inverse of the function , $f (x) = \sqrt[4]{6 x - 7}$

Short Answer

Expert verified

The inverse of, $f (x) = \sqrt[4]{6 x - 7}$ is

$f^{- 1} (x) = \frac{x^{4} + 7}{6}$

Step by step solution

Given information

Given the function $f (x) = \sqrt[4]{6 x - 7}$

Replace f(x) with y and then x,y

Replacing, we get

$f (x) = \sqrt[4]{6 x - 7} y = \sqrt[4]{6 x - 7} x = \sqrt[4]{6 y - 7} x^{4} = 6 y - 7 \frac{x^{4} + 7}{6} = y$

Substitute f-1(x) for y

Substituting, we get

$f^{- 1} (x) = \frac{x^{4} + 7}{6}$

Verify f(f-1(x))=x

Substituting, we get

$f (\frac{x^{4} + 7}{6}) = \sqrt[4]{6 (\frac{x^{4} + 7}{6}) - 7} f (\frac{x^{4} + 7}{6}) = \sqrt[4]{x^{4} + 7 - 7} f (\frac{x^{4} + 7}{6}) = \sqrt[4]{x^{4}} f (\frac{x^{4} + 7}{6}) = x$

Verify f-1(f(x))=x

Substituting, we get

$f^{- 1} (\sqrt[4]{6 x - 7}) = \sqrt[4]{6 (\frac{x^{4} + 7}{6}) - 7} f^{- 1} (\sqrt[4]{6 x - 7}) = \sqrt[4]{x^{4} + 7 - 7} f^{- 1} (\sqrt[4]{6 x - 7}) = \sqrt[4]{x^{4}} f^{- 1} (\sqrt[4]{6 x - 7}) = x$

So these are inverses

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Find the inverse of the function , $f (x) = \sqrt[4]{6 x - 7}$

Short Answer

Step by step solution

Given information

Replace f(x) with y and then x,y

Substitute f-1(x) for y

Verify f(f-1(x))=x

Verify f-1(f(x))=x

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Find the inverse of the function ,f(x)=6x-74

Short Answer

Step by step solution

Given information

Replace f(x) with y and then x,y

Substitute f-1(x) for y

Verify f(f-1(x))=x

Verify f-1(f(x))=x

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Study anywhere. Anytime. Across all devices.

Find the inverse of the function , $f (x) = \sqrt[4]{6 x - 7}$