If `f(x)=(2x+3)/(x-2)`, then: `(f@f)(x)=`

Options

A. `(3x+2)/(x-3)`
B. x
C. `(3x+4)/(x-3)`
D. none of these.

Answer

To find \( (f \circ f)(x) \) where \( f(x) = \frac{2x + 3}{x - 2} \), we need to substitute \( f(x) \) into itself. Here are the steps:

### Step 1: Write down the function
We have:
\[ f(x) = \frac{2x + 3}{x - 2} \]

### Step 2: Substitute \( f(x) \) into itself
We need to find \( f(f(x)) \). This means we will replace \( x \) in \( f(x) \) with \( f(x) \):
\[ f(f(x)) = f\left(\frac{2x + 3}{x - 2}\right) \]

### Step 3: Substitute into the function
Now, we substitute \( \frac{2x + 3}{x - 2} \) into the function:
\[ f\left(\frac{2x + 3}{x - 2}\right) = \frac{2\left(\frac{2x + 3}{x - 2}\right) + 3}{\left(\frac{2x + 3}{x - 2}\right) - 2} \]

### Step 4: Simplify the numerator
Calculating the numerator:
\[ 2\left(\frac{2x + 3}{x - 2}\right) + 3 = \frac{4x + 6}{x - 2} + 3 \]
To combine, we need a common denominator:
\[ = \frac{4x + 6 + 3(x - 2)}{x - 2} = \frac{4x + 6 + 3x - 6}{x - 2} = \frac{7x}{x - 2} \]

### Step 5: Simplify the denominator
Now, simplifying the denominator:
\[ \left(\frac{2x + 3}{x - 2}\right) - 2 = \frac{2x + 3 - 2(x - 2)}{x - 2} = \frac{2x + 3 - 2x + 4}{x - 2} = \frac{7}{x - 2} \]

### Step 6: Combine the results
Now we can combine the numerator and denominator:
\[ f(f(x)) = \frac{\frac{7x}{x - 2}}{\frac{7}{x - 2}} = \frac{7x}{7} = x \]

### Final Result
Thus, we have:
\[ (f \circ f)(x) = x \]