Let `f(x)=(5x+6)/(7x+9)` then, `f^-1(x)`=?

Explanation

To find the inverse of the function \( f(x) = \frac{5x + 6}{7x + 9} \), we will follow these steps:

### Step 1: Set \( f(x) \) equal to \( y \)
Let \( y = f(x) \). Therefore, we have:
\[ y = \frac{5x + 6}{7x + 9} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ y(7x + 9) = 5x + 6 \]

### Step 3: Distribute \( y \) on the left-hand side
Distributing \( y \) results in:
\[ 7xy + 9y = 5x + 6 \]

### Step 4: Rearrange the equation to isolate terms involving \( x \)
Rearranging the equation to group the terms involving \( x \) on one side gives:
\[ 7xy - 5x = 6 - 9y \]

### Step 5: Factor out \( x \) from the left-hand side
Factoring out \( x \) from the left side yields:
\[ x(7y - 5) = 6 - 9y \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by \( (7y - 5) \):
\[ x = \frac{6 - 9y}{7y - 5} \]

### Step 7: Write the inverse function
Since we have expressed \( x \) in terms of \( y \), we can write the inverse function:
\[ f^{-1}(y) = \frac{6 - 9y}{7y - 5} \]

### Step 8: Replace \( y \) with \( x \) to express the inverse function in standard notation
Thus, the inverse function is:
\[ f^{-1}(x) = \frac{6 - 9x}{7x - 5} \]

### Final Answer
The inverse of the function \( f(x) \) is:
\[ f^{-1}(x) = \frac{6 - 9x}{7x - 5} \]