Question

1. create a linear function g(x) that is perpendicular to the function and passes through the given value. f(x) = (2/3)x - 9 and f(4) = -2

Explanation:

Step1: Find the slope of g(x)

The slope of \( f(x)=\frac{2}{3}x - 9 \) is \( m_f=\frac{2}{3} \). For perpendicular lines, the slope \( m_g \) satisfies \( m_f\times m_g=- 1 \). So \( \frac{2}{3}\times m_g=-1 \), solving for \( m_g \) gives \( m_g =-\frac{3}{2} \).

Step2: Use point - slope form

The line \( g(x) \) passes through the point \( (4,-2) \) (since \( f(4) = - 2 \)). The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(4,-2) \) and \( m =-\frac{3}{2} \). Substituting these values: \( y-(-2)=-\frac{3}{2}(x - 4) \).

Step3: Simplify to slope - intercept form

Simplify the equation: \( y + 2=-\frac{3}{2}x+6 \). Subtract 2 from both sides: \( y=-\frac{3}{2}x + 4 \). So \( g(x)=-\frac{3}{2}x + 4 \).

Answer:

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