The function \( f \) is defined as \( f(x)=\frac{7}{2 x^{2}-4 x} \). Find \( f(x+6) \). Write your answer without parentheses, and simplify it as much as possible. \[ f(x+6)=\square \]

To find \( f(x+6) \), we substitute \( x+6 \) into the function \( f(x) \): \[ f(x) = \frac{7}{2x^2 - 4x} \] Now substituting \( x+6 \): \[ f(x+6) = \frac{7}{2(x+6)^2 - 4(x+6)} \] First, we will simplify the expression in the denominator: 1. Calculate \( (x+6)^2 = x^2 + 12x + 36 \). 2. Now, substitute this back into the denominator: \[ 2(x+6)^2 = 2(x^2 + 12x + 36) = 2x^2 + 24x + 72 \] \[ 4(x+6) = 4x + 24 \] Now combine them in the denominator: \[ f(x+6) = \frac{7}{(2x^2 + 24x + 72) - (4x + 24)} \] This simplifies to: \[ f(x+6) = \frac{7}{2x^2 + 24x + 72 - 4x - 24} \] So the expression becomes: \[ f(x+6) = \frac{7}{2x^2 + (24x - 4x) + (72 - 24)} = \frac{7}{2x^2 + 20x + 48} \] Thus, the simplified answer is: \[ f(x+6) = \frac{7}{2x^2 + 20x + 48} \]