Let f(x)= x-4/2x+3 then find f-1(2)

Answer 1

To find \( f^{-1}(2) \) for the function \( f(x) = \frac{x - 4}{2x + 3} \), we will first find the inverse function \( f^{-1}(x) \).

1. Start by letting \( y = f(x) \): \[ y = \frac{x - 4}{2x + 3} \]

2. Now, solve for \( x \) in terms of \( y \): \[ y(2x + 3) = x - 4 \]

   Expanding the left side: \[ 2xy + 3y = x - 4 \]

   Rearranging the equation: \[ x - 2xy = 3y + 4 \] \[ x(1 - 2y) = 3y + 4 \] \[ x = \frac{3y + 4}{1 - 2y} \]

3. Thus, the inverse function is: \[ f^{-1}(x) = \frac{3x + 4}{1 - 2x} \]

4. Now, we can find \( f^{-1}(2) \): \[ f^{-1}(2) = \frac{3(2) + 4}{1 - 2(2)} = \frac{6 + 4}{1 - 4} = \frac{10}{-3} = -\frac{10}{3} \]

So, the value of \( f^{-1}(2) \) is: \[ \boxed{-\frac{10}{3}} \]