Bernard Martin
11/19/2023 · High School
For the function \( f(x)=6 \sqrt[7]{x} \), find \( f^{-1}(x) \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the inverse function \( f^{-1}(x) \) for the function \( f(x) = 6 \sqrt[7]{x} \), we will follow these steps:
1. **Replace \( f(x) \) with \( y \)**:
\[
y = 6 \sqrt[7]{x}
\]
2. **Solve for \( x \) in terms of \( y \)**:
First, we isolate \( \sqrt[7]{x} \):
\[
\sqrt[7]{x} = \frac{y}{6}
\]
Next, we raise both sides to the power of 7 to eliminate the root:
\[
x = \left( \frac{y}{6} \right)^7
\]
3. **Express \( x \) as \( f^{-1}(y) \)**:
Now we can express \( x \) in terms of \( y \):
\[
f^{-1}(y) = \left( \frac{y}{6} \right)^7
\]
4. **Replace \( y \) with \( x \)** to write the inverse function:
\[
f^{-1}(x) = \left( \frac{x}{6} \right)^7
\]
Thus, the inverse function is:
\[
\boxed{f^{-1}(x) = \left( \frac{x}{6} \right)^7}
\]
Quick Answer
\( f^{-1}(x) = \left( \frac{x}{6} \right)^7 \)
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