If `f(x)=(3x+2)/(5x-3)`, then `f[f(x)]` is equal to:

Options

A. `-x`
B. `x`
C. 0
D. None of these

Answer

To find \( f[f(x)] \) where \( f(x) = \frac{3x + 2}{5x - 3} \), we will follow these steps:

### Step 1: Substitute \( f(x) \) into itself
We start by substituting \( f(x) \) into the function \( f \):
\[ f[f(x)] = f\left(\frac{3x + 2}{5x - 3}\right) \]
This means we replace \( x \) in \( f(x) \) with \( \frac{3x + 2}{5x - 3} \).

### Step 2: Write the expression for \( f(f(x)) \)
Now we will express \( f\left(\frac{3x + 2}{5x - 3}\right) \):
\[ f\left(\frac{3x + 2}{5x - 3}\right) = \frac{3\left(\frac{3x + 2}{5x - 3}\right) + 2}{5\left(\frac{3x + 2}{5x - 3}\right) - 3} \]

### Step 3: Simplify the numerator
Let's simplify the numerator:
\[ 3\left(\frac{3x + 2}{5x - 3}\right) + 2 = \frac{9x + 6}{5x - 3} + 2 = \frac{9x + 6 + 2(5x - 3)}{5x - 3} \]
Calculating \( 2(5x - 3) \):
\[ 2(5x - 3) = 10x - 6 \]
So the numerator becomes:
\[ 9x + 6 + 10x - 6 = 19x \]
Thus, the numerator simplifies to:
\[ \frac{19x}{5x - 3} \]

### Step 4: Simplify the denominator
Now, let's simplify the denominator:
\[ 5\left(\frac{3x + 2}{5x - 3}\right) - 3 = \frac{15x + 10}{5x - 3} - 3 = \frac{15x + 10 - 3(5x - 3)}{5x - 3} \]
Calculating \( -3(5x - 3) \):
\[ -3(5x - 3) = -15x + 9 \]
So the denominator becomes:
\[ 15x + 10 - 15x + 9 = 19 \]
Thus, the denominator simplifies to:
\[ \frac{19}{5x - 3} \]

### Step 5: Combine the results
Now we can combine the simplified numerator and denominator:
\[ f[f(x)] = \frac{\frac{19x}{5x - 3}}{\frac{19}{5x - 3}} = \frac{19x}{19} = x \]

### Final Result
Thus, we conclude that:
\[ f[f(x)] = x \]