Questions: Find the inverse function for f(x)=(-3x+4)/(1-4x) f^(-1)(x)=

Solution Steps

To find the inverse of the function \( f(x) = \frac{-3x + 4}{1 - 4x} \), we need to follow these steps:

1. Replace \( f(x) \) with \( y \).
2. Swap \( x \) and \( y \) to get the equation in terms of \( y \).
3. Solve the resulting equation for \( y \) to express it in terms of \( x \).
4. The expression for \( y \) is the inverse function \( f^{-1}(x) \).

We are given the function \( f(x) = \frac{-3x + 4}{1 - 4x} \) and need to find its inverse function, denoted as \( f^{-1}(x) \).

To find the inverse, we start by setting \( y = f(x) \). Thus, we have: \[ y = \frac{-3x + 4}{1 - 4x} \] We need to solve this equation for \( x \) in terms of \( y \).

To find the inverse, swap \( x \) and \( y \) in the equation: \[ x = \frac{-3y + 4}{1 - 4y} \]

Now, solve the equation for \( y \):

1. Multiply both sides by \( 1 - 4y \) to eliminate the fraction: \[ x(1 - 4y) = -3y + 4 \]
2. Distribute \( x \) on the left side: \[ x - 4xy = -3y + 4 \]
3. Rearrange the terms to isolate terms involving \( y \) on one side: \[ 4xy - 3y = x - 4 \]
4. Factor out \( y \) from the left side: \[ y(4x - 3) = x - 4 \]
5. Solve for \( y \): \[ y = \frac{x - 4}{4x - 3} \]

Final Answer