5. Show that f(x) and g(x) are inverse functions. and g(x)=√√8x f(x) = 2²
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![**Problem 5: Verify Inverse Functions**
**Task:** Show that \( f(x) \) and \( g(x) \) are inverse functions.
**Functions Given:**
- \( f(x) = \frac{x^{3}}{8} \)
- \( g(x) = \sqrt[3]{8x} \)
---
**To prove that \( f(x) \) and \( g(x) \) are inverse functions, you need to demonstrate the following:**
1. \( f(g(x)) = x \)
2. \( g(f(x)) = x \)
### Solution:
1. **Compute \( f(g(x)) \):**
- Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\sqrt[3]{8x}\right) = \frac{(\sqrt[3]{8x})^3}{8} = \frac{8x}{8} = x
\]
2. **Compute \( g(f(x)) \):**
- Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left(\frac{x^{3}}{8}\right) = \sqrt[3]{8 \times \frac{x^{3}}{8}} = \sqrt[3]{x^{3}} = x
\]
Since both compositions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, \( f(x) \) and \( g(x) \) are indeed inverse functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4339918c-b6fc-450e-92b7-d597aa3e9053%2F352f3c19-55b5-4748-a8ec-84944e6b015a%2F20n5t2b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5: Verify Inverse Functions**
**Task:** Show that \( f(x) \) and \( g(x) \) are inverse functions.
**Functions Given:**
- \( f(x) = \frac{x^{3}}{8} \)
- \( g(x) = \sqrt[3]{8x} \)
---
**To prove that \( f(x) \) and \( g(x) \) are inverse functions, you need to demonstrate the following:**
1. \( f(g(x)) = x \)
2. \( g(f(x)) = x \)
### Solution:
1. **Compute \( f(g(x)) \):**
- Substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\sqrt[3]{8x}\right) = \frac{(\sqrt[3]{8x})^3}{8} = \frac{8x}{8} = x
\]
2. **Compute \( g(f(x)) \):**
- Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g\left(\frac{x^{3}}{8}\right) = \sqrt[3]{8 \times \frac{x^{3}}{8}} = \sqrt[3]{x^{3}} = x
\]
Since both compositions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, \( f(x) \) and \( g(x) \) are indeed inverse functions.
Expert Solution
Step 1: First we find f^(-1)(x)
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