Questions: Find the inverse function of (f). (f(x)=frac1x+4) (f^-1(x)=)

Solution Steps

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

1. Replace \( f(x) \) with \( y \): \( y = \frac{1}{x+4} \).
2. Swap \( x \) and \( y \) to get \( x = \frac{1}{y+4} \).
3. Solve for \( y \) in terms of \( x \).

Given the function \( f(x) = \frac{1}{x+4} \), we start by replacing \( f(x) \) with \( y \): \[ y = \frac{1}{x+4} \]

Next, we swap \( x \) and \( y \) to find the inverse function: \[ x = \frac{1}{y+4} \]

We solve the equation \( x = \frac{1}{y+4} \) for \( y \): \[ x(y + 4) = 1 \implies xy + 4x = 1 \implies xy = 1 - 4x \implies y = \frac{1 - 4x}{x} \]

Final Answer