what is the inverse of $$ f\left(x\right)=\frac{8x+9}{7x-8} $$

To find the inverse of the function $ f(x) = \frac{8x + 9}{7x - 8} $, we follow these steps:

### Step 1: Replace $ f(x) $ with $ y $
We start by writing the equation as:
\[
y = \frac{8x + 9}{7x - 8}
\]

### Step 2: Swap $ x $ and $ y $
To find the inverse, swap $ x $ and $ y $:
\[
x = \frac{8y + 9}{7y - 8}
\]

### Step 3: Solve for $ y $
Now, solve for $ y $ in terms of $ x $. First, eliminate the denominator by multiplying both sides of the equation by $ 7y - 8 $:
\[
x(7y - 8) = 8y + 9
\]
Distribute the $ x $ on the left-hand side:
\[
7xy - 8x = 8y + 9
\]
Move all terms involving $ y $ to one side of the equation:
\[
7xy - 8y = 8x + 9
\]
Factor out $ y $ on the left-hand side:
\[
y(7x - 8) = 8x + 9
\]
Finally, solve for $ y $ by dividing both sides by $ 7x - 8 $:
\[
y = \frac{8x + 9}{7x - 8}
\]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{8x + 9}{7x - 8}
\]

### Conclusion:
The function $ f(x) = \frac{8x + 9}{7x - 8} $ is its own inverse. That means $ f^{-1}(x) = f(x) $.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What are the conditions under which a function is its own inverse?
2. How do we determine if a function is one-to-one (bijective)?
3. Can you find the inverse of a quadratic function?
4. How can you verify that two functions are inverses of each other?
5. What is the graphical interpretation of inverse functions?

**Tip:** When solving for the inverse of a rational function, always ensure to check if the function is one-to-one to guarantee the existence of an inverse!

What is the inverse of $$ f(x)=\frac{8x+9}{7x-8} $$?

Discover how to find the inverse of the rational function f(x) = (8x + 9)/(7x - 8). This detailed solution walks through key algebraic steps, including solving for the inverse and verifying that the function is its own inverse. Perfect 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