The one-to-one function \( g \) is defined below. \( g(x)=\frac{6 x-1}{x+4} \) Find \( g^{-1}(x) \), where \( g^{-1} \) is the inverse of \( g \). Also state the domain and range of \( g^{-1} \) in interval notation. \( g^{-1}(x)= \) Domain of \( g^{-1} \) : Range of \( g^{-1} \) :

To find the inverse of the function \( g(x) = \frac{6x - 1}{x + 4} \), we will follow these steps: 1. **Replace \( g(x) \) with \( y \)**: \[ y = \frac{6x - 1}{x + 4} \] 2. **Swap \( x \) and \( y \)** to find the inverse: \[ x = \frac{6y - 1}{y + 4} \] 3. **Solve for \( y \)**: - Multiply both sides by \( y + 4 \): \[ x(y + 4) = 6y - 1 \] - Distribute \( x \): \[ xy + 4x = 6y - 1 \] - Rearrange the equation to isolate terms involving \( y \): \[ xy - 6y = -1 - 4x \] - Factor out \( y \): \[ y(x - 6) = -1 - 4x \] - Solve for \( y \): \[ y = \frac{-1 - 4x}{x - 6} \] 4. **Thus, the inverse function is**: \[ g^{-1}(x) = \frac{-1 - 4x}{x - 6} \] Next, we will determine the domain and range of \( g^{-1}(x) \). ### Domain of \( g^{-1} \): The domain of \( g^{-1}(x) \) is determined by the values of \( x \) that do not make the denominator zero: \[ x - 6 \neq 0 \implies x \neq 6 \] Thus, the domain of \( g^{-1} \) in interval notation is: \[ (-\infty, 6) \cup (6, \infty) \] ### Range of \( g^{-1} \): The range of \( g^{-1}(x) \) corresponds to the domain of \( g(x) \). The function \( g(x) \) is a rational function, and we need to find its domain: The denominator \( x + 4 \) cannot be zero: \[ x + 4 \neq 0 \implies x \neq -4 \] Thus, the domain of \( g(x) \) is: \[ (-\infty, -4) \cup (-4, \infty) \] This means the range of \( g^{-1}(x) \) is: \[ (-\infty, -4) \cup (-4, \infty) \] ### Final Answer: \[ g^{-1}(x) = \frac{-1 - 4x}{x - 6} \] Domain of \( g^{-1} \): \( (-\infty, 6) \cup (6, \infty) \) Range of \( g^{-1} \): \( (-\infty, -4) \cup (-4, \infty) \)