Find the inverse of f(x) f-¹(x) = = - 3x 4x - 2 1 -

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem:** Find the inverse of the function \( f(x) = \frac{-3x - 1}{4x - 2} \).

**Solution:** 
To find the inverse of a function, we essentially want to solve for \( x \) in terms of \( y \) and then swap \( x \) and \( y \).

1. Start with the equation for the function:
   \[
   y = \frac{-3x - 1}{4x - 2}
   \]

2. Swap \( y \) and \( x \):
   \[
   x = \frac{-3y - 1}{4y - 2}
   \]

3. Multiply both sides by \( (4y - 2) \) to eliminate the denominator:
   \[
   x(4y - 2) = -3y - 1
   \]

4. Distribute \( x \):
   \[
   4xy - 2x = -3y - 1
   \]

5. Rearrange the terms to gather all terms involving \( y \) on one side:
   \[
   4xy + 3y = 2x - 1
   \]

6. Factor out \( y \) on the left side:
   \[
   y(4x + 3) = 2x - 1
   \]

7. Solve for \( y \):
   \[
   y = \frac{2x - 1}{4x + 3}
   \]

Therefore, the inverse function is:
\[
f^{-1}(x) = \frac{2x - 1}{4x + 3}
\]
Transcribed Image Text:**Problem:** Find the inverse of the function \( f(x) = \frac{-3x - 1}{4x - 2} \). **Solution:** To find the inverse of a function, we essentially want to solve for \( x \) in terms of \( y \) and then swap \( x \) and \( y \). 1. Start with the equation for the function: \[ y = \frac{-3x - 1}{4x - 2} \] 2. Swap \( y \) and \( x \): \[ x = \frac{-3y - 1}{4y - 2} \] 3. Multiply both sides by \( (4y - 2) \) to eliminate the denominator: \[ x(4y - 2) = -3y - 1 \] 4. Distribute \( x \): \[ 4xy - 2x = -3y - 1 \] 5. Rearrange the terms to gather all terms involving \( y \) on one side: \[ 4xy + 3y = 2x - 1 \] 6. Factor out \( y \) on the left side: \[ y(4x + 3) = 2x - 1 \] 7. Solve for \( y \): \[ y = \frac{2x - 1}{4x + 3} \] Therefore, the inverse function is: \[ f^{-1}(x) = \frac{2x - 1}{4x + 3} \]
Given that \( f(x) = 9x + 5 \) and \( g(x) = 3 - x^2 \), calculate:

(a) \( f(g(0)) = \_\_\_\_\_\_\_\_\_\_ \)

(b) \( g(f(0)) = \_\_\_\_\_\_\_\_\_\_ \)
Transcribed Image Text:Given that \( f(x) = 9x + 5 \) and \( g(x) = 3 - x^2 \), calculate: (a) \( f(g(0)) = \_\_\_\_\_\_\_\_\_\_ \) (b) \( g(f(0)) = \_\_\_\_\_\_\_\_\_\_ \)
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