We are given two functions $f(x) = \frac{2x+7}{7}$ and $g(x) = \frac{3x-6}{6}$. We need to find $g(6)$ and $f^{-1}(x)$.

1. Problem Description

We are given two functions $f(x) = \frac{2x+7}{7}$ and $g(x) = \frac{3x-6}{6}$.

We need to find $g(6)$ and $f^{-1}(x)$.

2. Solution Steps

a. To find $g(6)$, we substitute $x=6$ into the function $g(x)$.

$g(x) = \frac{3x-6}{6}$

$g(6) = \frac{3(6)-6}{6} = \frac{18-6}{6} = \frac{12}{6} = 2$

b. To find the inverse of $f(x)$, denoted as $f^{-1}(x)$, we let $y = f(x)$ and solve for $x$ in terms of $y$.

$y = \frac{2x+7}{7}$

Multiply both sides by 7:

$7y = 2x+7$

Subtract 7 from both sides:

$7y - 7 = 2x$

Divide both sides by 2:

$x = \frac{7y-7}{2}$

Now we swap $x$ and $y$ to get the inverse function:

$y = \frac{7x-7}{2}$

Therefore, $f^{-1}(x) = \frac{7x-7}{2}$.

3. Final Answer

a. $g(6) = 2$

b. $f^{-1}(x) = \frac{7x-7}{2}$