the function f(x)=\sqrt[4]{7x} is one to one. find an equation for f^{-1}\left(x\right), the inverse function

To find the inverse function \( f^{-1}(x) \) of \( f(x) = \sqrt[4]{7x} \), follow these steps:

1. **Express the function in terms of \( y \):**
   \[
   y = \sqrt[4]{7x}
   \]

2. **Solve for \( x \) in terms of \( y \):**
   To eliminate the fourth root, raise both sides to the power of 4:
   \[
   y^4 = 7x
   \]
   Now, solve for \( x \):
   \[
   x = \frac{y^4}{7}
   \]

3. **Interchange \( x \) and \( y \):**
   Since we are looking for the inverse function, switch \( x \) and \( y \):
   \[
   y = \frac{x^4}{7}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{x^4}{7}
\]

### Do you have any questions or want further details?

Here are 5 related questions to explore:
1. How do you verify if a function is one-to-one?
2. What is the significance of finding the inverse of a function?
3. Can all functions have inverses?
4. How do domain restrictions affect the inverse function?
5. How would the process change if the function had a different root or power?

**Tip:** To check if two functions are truly inverses, apply both compositions \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \). They should return \( x \).

The function f(x)=\sqrt[4]{7x} is one-to-one. Find an equation for f^{-1}(x), the inverse function.

Discover how to find the inverse function of f(x)=\sqrt[4]{7x} with a clear, detailed explanation. This problem focuses on inverse functions, radical functions, and the algebraic process for solving one-to-one functions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation