If `f(x)=(2x+3)/(3x-2),xne(2)/(3)`, then find fof (x).

Answer

To find \( f(f(x)) \) where \( f(x) = \frac{2x + 3}{3x - 2} \) and \( x \neq \frac{2}{3} \), we will follow these steps:

### Step 1: Substitute \( f(x) \) into itself
We need to calculate \( f(f(x)) \). This means we will substitute \( f(x) \) into the function \( f \).

\[ f(f(x)) = f\left(\frac{2x + 3}{3x - 2}\right) \]

### Step 2: Replace \( x \) in \( f(x) \)
Now we will replace \( x \) in the function \( f(x) \) with \( \frac{2x + 3}{3x - 2} \):

\[ f\left(\frac{2x + 3}{3x - 2}\right) = \frac{2\left(\frac{2x + 3}{3x - 2}\right) + 3}{3\left(\frac{2x + 3}{3x - 2}\right) - 2} \]

### Step 3: Simplify the numerator
First, simplify the numerator:

\[ 2\left(\frac{2x + 3}{3x - 2}\right) + 3 = \frac{4x + 6}{3x - 2} + 3 \]

To combine these fractions, we need a common denominator:

\[ = \frac{4x + 6 + 3(3x - 2)}{3x - 2} = \frac{4x + 6 + 9x - 6}{3x - 2} = \frac{13x}{3x - 2} \]

### Step 4: Simplify the denominator
Now simplify the denominator:

\[ 3\left(\frac{2x + 3}{3x - 2}\right) - 2 = \frac{6x + 9}{3x - 2} - 2 \]

Again, we need a common denominator:

\[ = \frac{6x + 9 - 2(3x - 2)}{3x - 2} = \frac{6x + 9 - 6x + 4}{3x - 2} = \frac{13}{3x - 2} \]

### Step 5: Combine the results
Now we can combine the simplified numerator and denominator:

\[ f(f(x)) = \frac{\frac{13x}{3x - 2}}{\frac{13}{3x - 2}} = \frac{13x}{13} = x \]

### Final Answer
Thus, we find that:

\[ f(f(x)) = x \]