### Understanding Functions and Their Inverses The function \( f(x) = \frac{8}{9 + x} \) is one-to-one. #### Task: (a) Find its inverse and check your answer. (b) Find the domain and the range of \( f \) and \( f^{-1} \). ### Step-by-Step Solution: #### Part (a) - Finding the Inverse \[ f^{-1}(x) = \boxed{} \] (Simplify your answer.) #### Part (b) - Finding the Domain and Range **Find the domain of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** - \( \bigcirc \) A. The domain is \( \{ x | x \ne \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) B. The domain is \( \{ x | x \le \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) C. The domain is \( \{ x | x \ge \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) D. The domain is the set of all real numbers. **Find the range of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** - \( \bigcirc \) A. The range is \( \{ y | y \ge \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) B. The range is \( \{ y | y \ne \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) Click to select your answer(s).
### Understanding Functions and Their Inverses The function \( f(x) = \frac{8}{9 + x} \) is one-to-one. #### Task: (a) Find its inverse and check your answer. (b) Find the domain and the range of \( f \) and \( f^{-1} \). ### Step-by-Step Solution: #### Part (a) - Finding the Inverse \[ f^{-1}(x) = \boxed{} \] (Simplify your answer.) #### Part (b) - Finding the Domain and Range **Find the domain of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** - \( \bigcirc \) A. The domain is \( \{ x | x \ne \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) B. The domain is \( \{ x | x \le \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) C. The domain is \( \{ x | x \ge \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) D. The domain is the set of all real numbers. **Find the range of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** - \( \bigcirc \) A. The range is \( \{ y | y \ge \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) - \( \bigcirc \) B. The range is \( \{ y | y \ne \boxed{} \} \). (Type integers or fractions. Use a comma to separate answers as needed.) Click to select your answer(s).
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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![### Understanding Functions and Their Inverses
The function \( f(x) = \frac{8}{9 + x} \) is one-to-one.
#### Task:
(a) Find its inverse and check your answer.
(b) Find the domain and the range of \( f \) and \( f^{-1} \).
### Step-by-Step Solution:
#### Part (a) - Finding the Inverse
\[ f^{-1}(x) = \boxed{} \]
(Simplify your answer.)
#### Part (b) - Finding the Domain and Range
**Find the domain of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.**
- \( \bigcirc \) A. The domain is \( \{ x | x \ne \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) B. The domain is \( \{ x | x \le \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) C. The domain is \( \{ x | x \ge \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) D. The domain is the set of all real numbers.
**Find the range of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.**
- \( \bigcirc \) A. The range is \( \{ y | y \ge \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) B. The range is \( \{ y | y \ne \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
Click to select your answer(s).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6811bcd-02f8-46fa-b044-8f44b9c4930d%2F05cc64c0-e377-45fd-8237-4c5380360a76%2F1utel0e.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Functions and Their Inverses
The function \( f(x) = \frac{8}{9 + x} \) is one-to-one.
#### Task:
(a) Find its inverse and check your answer.
(b) Find the domain and the range of \( f \) and \( f^{-1} \).
### Step-by-Step Solution:
#### Part (a) - Finding the Inverse
\[ f^{-1}(x) = \boxed{} \]
(Simplify your answer.)
#### Part (b) - Finding the Domain and Range
**Find the domain of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.**
- \( \bigcirc \) A. The domain is \( \{ x | x \ne \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) B. The domain is \( \{ x | x \le \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) C. The domain is \( \{ x | x \ge \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) D. The domain is the set of all real numbers.
**Find the range of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.**
- \( \bigcirc \) A. The range is \( \{ y | y \ge \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
- \( \bigcirc \) B. The range is \( \{ y | y \ne \boxed{} \} \).
(Type integers or fractions. Use a comma to separate answers as needed.)
Click to select your answer(s).
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