Q. 71 the function f is one-to-one. Fi... [FREE SOLUTION]

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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 71 (page 269)

the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{x^{2} - 4}{2 x^{2}} x > 0$

Short Answer

Expert verified

The inverse of the function $f (x) = \frac{x^{2} - 4}{2 x^{2}}$ is $f^{- 1} (x) = \sqrt{\frac{- 4}{2 x - 1}}, x > \frac{1}{2}$

Step by step solution

Step 1. Given data

The given function is

$f (x) = \frac{x^{2} - 4}{2 x^{2}}$

Step 2. interchanging variables

Replace with y and interchange x and y

$f (x) = \frac{x^{2} - 4}{2 x^{2}} y = \frac{x^{2} - 4}{2 x^{2}} x = \frac{y^{2} - 4}{2 y^{2}}$

Step 3. The inverse of the function

Solve the equation for y

$x = \frac{y^{2} - 4}{2 y^{2}} x (2 y^{2}) = y^{2} - 4 2 x y^{2} = y^{2} - 4 2 x y^{2} - y^{2} = - 4 y^{2} (2 x - 1) = - 4 y^{2} = \frac{- 4}{2 x - 1} y = \sqrt{\frac{- 4}{2 x - 1}}$

replace ywith $f^{- 1} (x)$

role="math" localid="1646161961147" $f^{- 1} (x) = \sqrt{\frac{- 4}{2 x - 1}}$

Step 4. Verification

Determine $f (f^{- 1} (x)) = f (\sqrt{\frac{- 4}{2 x - 1}}) f (f^{- 1} (x)) = \frac{{(\sqrt{\frac{- 4}{2 x - 1}})}^{2} - 4}{2 {(\sqrt{\frac{- 4}{2 x - 1}})}^{2}} f (f^{- 1} (x)) = \frac{- 4 - 4 (2 x - 1)}{- 8} f (f^{- 1} (x)) = \frac{- 4 - 8 x - 4}{- 8} f (f^{- 1} (x)) = \frac{- 8 x}{- 8} f (f^{- 1} (x)) = x$

Determine $f^{- 1} (f (x))$

$f^{- 1} (f (x)) = f^{- 1} (\frac{x^{2} - 4}{2 x^{2}}) f^{- 1} (f (x)) = \sqrt{\frac{- 4}{2 (\frac{x^{2} - 4}{2 x^{2}}) - 1}} f^{- 1} (f (x)) = \sqrt{\frac{- 4 x^{2}}{x^{2} - 4 - x^{2}}} f^{- 1} (f (x)) = \sqrt{\frac{- 4 x^{2}}{- 4}} f^{- 1} (f (x)) = \sqrt{x^{2}} f^{- 1} (f (x)) = x$

$f^{- 1} (f (x)) = x & f (f^{- 1} (x)) = x$ so inverse function is correct

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91影视

the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{x^{2} - 4}{2 x^{2}} x > 0$

Short Answer

Step by step solution

Step 1. Given data

Step 2. interchanging variables

Step 3. The inverse of the function

Step 4. Verification

One App. One Place for Learning.

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91影视

the function f is one-to-one. Find its inverse and check your answer.f(x)=x2-42x2x>0

Short Answer

Step by step solution

Step 1. Given data

Step 2. interchanging variables

Step 3. The inverse of the function

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Logic and Functions

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

the function f is one-to-one. Find its inverse and check your answer.
$f (x) = \frac{x^{2} - 4}{2 x^{2}} x > 0$