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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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

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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

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

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