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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Question
![**Problem Statement:**
If \( f(x) = \sqrt{7 - 4x} \), find a formula for \( f^{-1}(x) \).
**Solution Approach:**
1. Start by replacing \( f(x) \) with \( y \) for simplicity:
\[
y = \sqrt{7 - 4x}
\]
2. Solve for \( x \) in terms of \( y \):
- Square both sides to remove the square root:
\[
y^2 = 7 - 4x
\]
- Rearrange to solve for \( x \):
\[
4x = 7 - y^2
\]
\[
x = \frac{7 - y^2}{4}
\]
3. Interchange \( x \) and \( y \) to find \( f^{-1}(x) \):
\[
y = \frac{7 - x^2}{4}
\]
Thus, the inverse function is:
\[
f^{-1}(x) = \frac{7 - x^2}{4}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F220673f3-4665-41cf-82e7-eaf0663df6ee%2F8eefc55f-c2a1-41ab-9315-db6ae6b4cf66%2Fo3yhkh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( f(x) = \sqrt{7 - 4x} \), find a formula for \( f^{-1}(x) \).
**Solution Approach:**
1. Start by replacing \( f(x) \) with \( y \) for simplicity:
\[
y = \sqrt{7 - 4x}
\]
2. Solve for \( x \) in terms of \( y \):
- Square both sides to remove the square root:
\[
y^2 = 7 - 4x
\]
- Rearrange to solve for \( x \):
\[
4x = 7 - y^2
\]
\[
x = \frac{7 - y^2}{4}
\]
3. Interchange \( x \) and \( y \) to find \( f^{-1}(x) \):
\[
y = \frac{7 - x^2}{4}
\]
Thus, the inverse function is:
\[
f^{-1}(x) = \frac{7 - x^2}{4}
\]
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