https://file.aiquickdraw.com/mathbot/file/1729019780522wube5x8v.jpg

To find the inverse of the given function \( f(x) = \frac{x - 6}{3x - 4} \), we will follow these steps:

### Step 1: Replace \( f(x) \) with \( y \).
\[
y = \frac{x - 6}{3x - 4}
\]

### Step 2: Swap \( x \) and \( y \).
\[
x = \frac{y - 6}{3y - 4}
\]

### Step 3: Solve for \( y \).
To isolate \( y \), multiply both sides by \( 3y - 4 \):
\[
x(3y - 4) = y - 6
\]
Expanding this gives:
\[
3xy - 4x = y - 6
\]

### Step 4: Rearrange the equation.
Bring all terms involving \( y \) to one side:
\[
3xy - y = 4x - 6
\]

Factor out \( y \):
\[
y(3x - 1) = 4x - 6
\]

### Step 5: Solve for \( y \).
\[
y = \frac{4x - 6}{3x - 1}
\]

### Step 6: Write the inverse function.
\[
f^{-1}(x) = \frac{4x - 6}{3x - 1}
\]

The inverse function of \( f(x) \) is \( f^{-1}(x) = \frac{4x - 6}{3x - 1} \).

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you verify if two functions are inverses of each other?
2. What is the domain of the function \( f(x) \) and its inverse \( f^{-1}(x) \)?
3. How do you find the inverse of a quadratic function?
4. What happens to the graph of \( f(x) \) when you find its inverse?
5. How do you determine if a function is one-to-one?

**Tip:** To find the inverse of a function, always start by swapping \( x \) and \( y \), then solve for \( y \).

Find the inverse function of f(x) = (x - 6) / (3x - 4).

Learn how to find the inverse of a rational function, f(x) = (x - 6) / (3x - 4). This problem involves algebraic manipulation, solving for the inverse function, and verifying the result. Suitable for 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