To find the inverse of the function \( f(x) = \frac{2x - 3}{x - 4} \), we will follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = \frac{2x - 3}{x - 4}
\]

### Step 2: Swap \( x \) and \( y \) to find the inverse
\[
x = \frac{2y - 3}{y - 4}
\]

### Step 3: Solve for \( y \)

First, multiply both sides by \( y - 4 \) to eliminate the denominator on the right side:
\[
x(y - 4) = 2y - 3
\]
Expand both sides:
\[
xy - 4x = 2y - 3
\]
Move all terms involving \( y \) to one side:
\[
xy - 2y = 4x - 3
\]
Factor out \( y \) on the left-hand side:
\[
y(x - 2) = 4x - 3
\]
Now, solve for \( y \) by dividing both sides by \( x - 2 \):
\[
y = \frac{4x - 3}{x - 2}
\]

### Final Answer:
The inverse function is:
\[
f^{-1}(x) = \frac{4x - 3}{x - 2}
\]

Would you like further details or have any questions?

Here are five related questions for further exploration:
1. How do you verify that two functions are inverses of each other?
2. What are the domain and range of the inverse function \( f^{-1}(x) \)?
3. What happens if the original function \( f(x) \) has restrictions on its domain?
4. How do you find the inverse of more complex rational functions?
5. Can a function have more than one inverse?

**Tip:** Always check the domain of the original function before finding the inverse, as it may affect the range of the inverse function.

Find the inverse function of (2x-3)/(x-4).

Explore a step-by-step solution to find the inverse of the rational function (2x-3)/(x-4). This problem introduces algebraic concepts like variable swapping, rational functions, and the inverse function property, 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