https://file.aiquickdraw.com/mathbot/file/17269824800960nr80gv6.jpg

We are asked to find the inverse of the function \( f(x) = 6 + \sqrt{6x - 4} \), specifically \( f^{-1}(x) \).

### Steps to Find \( f^{-1}(x) \):

1. **Rewrite the equation** for \( y = f(x) \):
   \[
   y = 6 + \sqrt{6x - 4}
   \]

2. **Solve for \( x \) in terms of \( y \)**:
   - Subtract 6 from both sides:
     \[
     y - 6 = \sqrt{6x - 4}
     \]
   - Square both sides to remove the square root:
     \[
     (y - 6)^2 = 6x - 4
     \]
   - Add 4 to both sides:
     \[
     (y - 6)^2 + 4 = 6x
     \]
   - Divide by 6:
     \[
     x = \frac{(y - 6)^2 + 4}{6}
     \]

3. **Replace \( y \) with \( x \)** to express the inverse function:
   \[
   f^{-1}(x) = \frac{(x - 6)^2 + 4}{6}
   \]

Thus, the inverse function is:

\[
f^{-1}(x) = \frac{(x - 6)^2 + 4}{6}, \quad \text{for} \ x \geq 6
\]

Would you like further explanation on any of these steps?

Here are 5 related questions for further exploration:
1. What are the conditions for a function to have an inverse?
2. How can you verify if a function and its inverse are correct?
3. What role does the domain of \( f(x) \) play in determining \( f^{-1}(x) \)?
4. How would you find the derivative of \( f^{-1}(x) \)?
5. Can every function be inverted? Why or why not?

**Tip:** Always check the domain of the original function before solving for the inverse. This ensures that the inverse function's domain is valid and matches the original function's behavior.

Let f(x) = 6 + √(6x - 4). Find f^{-1}(x).

Learn how to find the inverse function of f(x) = 6 + √(6x - 4) with a step-by-step solution. This algebra problem demonstrates inverse functions, solving equations, and working with square roots. 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