Questions: If f(x) is a linear function, f(-4)=-5, and f(4)=4, find an equation for f(x) f(x)=

Find the equation of the linear function \( f(x) \) given \( f(-4) = -5 \) and \( f(4) = 4 \).

Step 1: Determine the slope of the linear function.

The slope \( m \) of a linear function passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the given points \( (-4, -5) \) and \( (4, 4) \): \[ m = \frac{4 - (-5)}{4 - (-4)} = \frac{9}{8} \]

Step 2: Use the point-slope form to find the equation of the line.

The point-slope form of a linear equation is: \[ y - y_1 = m(x - x_1) \] Using the slope \( m = \frac{9}{8} \) and the point \( (4, 4) \): \[ y - 4 = \frac{9}{8}(x - 4) \] Simplify the equation: \[ y = \frac{9}{8}x - \frac{9}{8} \cdot 4 + 4 \] \[ y = \frac{9}{8}x - \frac{36}{8} + 4 \] \[ y = \frac{9}{8}x - \frac{9}{2} + 4 \] \[ y = \frac{9}{8}x - \frac{1}{2} \]

The equation of the linear function is: \[ \boxed{f(x) = \frac{9}{8}x - \frac{1}{2}} \]

The equation of the linear function is: \[ \boxed{f(x) = \frac{9}{8}x - \frac{1}{2}} \]