Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Determine whether the two functions \( g(x) = (x - 5)^3 \) and \( h(x) = \sqrt[3]{x} + 5 \) are inverses.**
### Explanation
To determine if two functions are inverses, you can verify if their composition results in the identity function. Specifically, if \( g(h(x)) = x \) and \( h(g(x)) = x \) for all \( x \) in the domains, then the functions are inverses of each other.
#### Step-by-Step Guide
1. **Compose \( g(h(x)) \):**
- Substitute \( h(x) \) into \( g(x) \): \( g(h(x)) = g(\sqrt[3]{x} + 5) \).
- Simplify: \( g(\sqrt[3]{x} + 5) = (\sqrt[3]{x} + 5 - 5)^3 = (\sqrt[3]{x})^3 = x \).
2. **Compose \( h(g(x)) \):**
- Substitute \( g(x) \) into \( h(x) \): \( h(g(x)) = h((x - 5)^3) \).
- Simplify: \( h((x - 5)^3) = \sqrt[3]{(x - 5)^3} + 5 = x - 5 + 5 = x \).
Since both compositions \( g(h(x)) \) and \( h(g(x)) \) simplify to \( x \), the functions \( g(x) \) and \( h(x) \) are indeed inverses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93b779eb-3fb4-4e8e-a3e5-cbc97ed5211a%2Fdf0ed9a1-8d3c-4edd-895a-e5fd467e138a%2Fyh06gan_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the two functions \( g(x) = (x - 5)^3 \) and \( h(x) = \sqrt[3]{x} + 5 \) are inverses.**
### Explanation
To determine if two functions are inverses, you can verify if their composition results in the identity function. Specifically, if \( g(h(x)) = x \) and \( h(g(x)) = x \) for all \( x \) in the domains, then the functions are inverses of each other.
#### Step-by-Step Guide
1. **Compose \( g(h(x)) \):**
- Substitute \( h(x) \) into \( g(x) \): \( g(h(x)) = g(\sqrt[3]{x} + 5) \).
- Simplify: \( g(\sqrt[3]{x} + 5) = (\sqrt[3]{x} + 5 - 5)^3 = (\sqrt[3]{x})^3 = x \).
2. **Compose \( h(g(x)) \):**
- Substitute \( g(x) \) into \( h(x) \): \( h(g(x)) = h((x - 5)^3) \).
- Simplify: \( h((x - 5)^3) = \sqrt[3]{(x - 5)^3} + 5 = x - 5 + 5 = x \).
Since both compositions \( g(h(x)) \) and \( h(g(x)) \) simplify to \( x \), the functions \( g(x) \) and \( h(x) \) are indeed inverses.
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