If g(x)= 𝑥𝑥+1x+1x​ then the inverse function g - 1(x) =A.𝑥+1𝑥xx+1​ B.xC.𝑥1−𝑥1−xx​ D.𝑥−1𝑥xx−1​ E.𝑥𝑥+1x+1x​ SUBMITarrow_backPREVIOUS

To find the inverse function $g^{-1}(x)$, let's follow the steps outlined:

1. Break Down the Problem

We are given the function: $g(x) = \frac{x}{x + 1}$ We need to determine its inverse, $g^{-1}(x)$.

2. Relevant Concepts

To find the inverse of a function, we typically do the following:

• Swap $x$ and $y$ in the equation.
• Solve for $y$.

3. Analysis and Detail

1. Start with the equation: $y = g(x) = \frac{x}{x + 1}$

2. Swap $x$ and $y$: $x = \frac{y}{y + 1}$

3. Multiply both sides by $(y + 1)$ to eliminate the fraction: $x(y + 1) = y$ $xy + x = y$

4. Rearranging gives: $xy - y = -x$ $y(x - 1) = -x$

5. Finally, solve for $y$: $y = \frac{-x}{x - 1}$ This simplifies to: $y = \frac{x}{1 - x}$

4. Verify and Summarize

Thus, the inverse function is: $g^{-1}(x) = \frac{x}{1 - x}$

Final Answer

The inverse function $g^{-1}(x)$ is: $\frac{x}{1 - x}$ This corresponds to option C.