King Bond
09/23/2023 · Middle School
Which function is the inverse of \( g(x)=2 \sqrt[3]{x-3}+4 \) ? \( f(x)=\frac{(x-4)^{3}}{8}+3 \) \( f(x)=\frac{(x-2)^{3}}{64}+3 \) \( f(x)=\frac{(x-4)^{3}}{2}+3 \) \( f(x)=\frac{(x-2)^{3}}{4}+3 \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the inverse of the function \( g(x) = 2 \sqrt[3]{x-3} + 4 \), we will follow these steps:
1. Replace \( g(x) \) with \( y \):
\[
y = 2 \sqrt[3]{x-3} + 4
\]
2. Solve for \( x \) in terms of \( y \):
\[
y - 4 = 2 \sqrt[3]{x-3}
\]
\[
\frac{y - 4}{2} = \sqrt[3]{x-3}
\]
Now, cube both sides:
\[
\left( \frac{y - 4}{2} \right)^3 = x - 3
\]
\[
x = \left( \frac{y - 4}{2} \right)^3 + 3
\]
3. Now, replace \( y \) with \( x \) to express the inverse function:
\[
g^{-1}(x) = \left( \frac{x - 4}{2} \right)^3 + 3
\]
4. Simplify the expression:
\[
g^{-1}(x) = \frac{(x - 4)^3}{8} + 3
\]
Now, we can compare this result with the given options:
- \( f(x) = \frac{(x-4)^{3}}{8}+3 \)
- \( f(x) = \frac{(x-2)^{3}}{64}+3 \)
- \( f(x) = \frac{(x-4)^{3}}{2}+3 \)
- \( f(x) = \frac{(x-2)^{3}}{4}+3 \)
The correct inverse function is:
\[
\boxed{f(x) = \frac{(x-4)^{3}}{8}+3}
\]
Quick Answer
The inverse function is \( f(x) = \frac{(x-4)^{3}}{8}+3 \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
(0)
Ask Real Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
PREV 
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Uploaded Files
xxxx.png0%
Submit
