Question

1. create a linear function g(x) that is perpendicular to the function and passes through the given value.

$f(x) = \frac{2}{3}x - 9$ and $f(4) = -2$

Explanation:

Step1: Find slope of g(x)

For perpendicular lines, slopes are negative reciprocals. Slope of \( f(x) = \frac{2}{3}x - 9 \) is \( m_f = \frac{2}{3} \). So slope of \( g(x) \), \( m_g = -\frac{3}{2} \).

Step2: Identify point for g(x)

Given \( f(4) = -2 \), so the point \( (4, -2) \) lies on \( g(x) \).

Step3: Use point - slope form

Point - slope form: \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(4, - 2) \) and \( m = -\frac{3}{2} \).
\( y - (-2)=-\frac{3}{2}(x - 4) \)
\( y + 2=-\frac{3}{2}x+6 \)
\( y=-\frac{3}{2}x + 4 \)
So \( g(x)=-\frac{3}{2}x + 4 \)

Answer:

\( g(x)=-\frac{3}{2}x + 4 \)