To determine the slope of line \( g \), we first need to find the slope of line \( f \). The slope \( m \) of a line that passes through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For line \( f \), the points are \((2, 2)\) and \((4, 9)\).
Plugging in the points:
\[ m_f = \frac{9 - 2}{4 - 2} = \frac{7}{2} \]
So, the slope of line \( f \) is \(\frac{7}{2}\).
Since line \( g \) is perpendicular to line \( f\), the slope of line \( g \) (denoted as \( m_g \)) is the negative reciprocal of the slope of line \( f \). The negative reciprocal is found by taking the negative of the reciprocal of \( m_f \):
\[ m_g = -\frac{1}{m_f} = -\frac{1}{\frac{7}{2}} = -\frac{2}{7} \]
Thus, the slope of line \( g \) is
\[ \boxed{-\frac{2}{7}}. \]