Questions: Find the inverse of the following function, if possible. g(x) = 3/(2x+4) g^(-1)(x) =

Solution Steps

To find the inverse of the function \( g(x) = \frac{3}{2x + 4} \), we need to solve for \( x \) in terms of \( y \) where \( y = g(x) \). This involves swapping \( x \) and \( y \) and then solving the resulting equation for \( y \).

We start with the function given by

\[ g(x) = \frac{3}{2x + 4} \]

To find the inverse, we set \( y = g(x) \):

\[ y = \frac{3}{2x + 4} \]

Next, we rearrange the equation to solve for \( x \):

1. Multiply both sides by \( 2x + 4 \):

\[ y(2x + 4) = 3 \]

2. Expand and rearrange:

\[ 2xy + 4y = 3 \]

3. Isolate the term with \( x \):

\[ 2xy = 3 - 4y \]

4. Finally, solve for \( x \):

\[ x = -2 + \frac{3}{2y} \]

Thus, the inverse function \( g^{-1}(x) \) is given by:

\[ g^{-1}(x) = -2 + \frac{3}{2x} \]

Final Answer