Questions: A one-to-one function is given. Write an equation for the inverse function. g(x) = (8-x)/9 g^-1(x) =

Solution Steps

To find the inverse of the function \( g(x) = \frac{8-x}{9} \), we need to follow these steps:

1. Replace \( g(x) \) with \( y \).
2. Swap \( x \) and \( y \) to reflect the inverse relationship.
3. Solve for \( y \) in terms of \( x \).
4. Replace \( y \) with \( g^{-1}(x) \).

We start with the function defined as: \[ g(x) = \frac{8 - x}{9} \]

We set: \[ y = \frac{8 - x}{9} \]

To find the inverse, we swap \( x \) and \( y \): \[ x = \frac{8 - y}{9} \]

Now, we solve for \( y \): \[ 9x = 8 - y \] Rearranging gives: \[ y = 8 - 9x \]

Thus, the inverse function is: \[ g^{-1}(x) = 8 - 9x \]

Final Answer