Let's solve this step-by-step:

Problem Setup

We are given two points on the function $f(x)$:

• $f(3) = 2$, meaning the point $(3, 2)$.
• $f(6) = -3$, meaning the point $(6, -3)$.

The task is to determine which of the four given options describes $f(x)$.

Step 1: Find the Slope

The slope $m$ of the line passing through the points $(x_1, y_1) = (3, 2)$ and $(x_2, y_2) = (6, -3)$ is calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{6 - 3} = \frac{-5}{3}.$

Step 2: Use the Point-Slope Formula

The point-slope form of a linear equation is: $y - y_1 = m(x - x_1).$ Substituting $m = -\frac{5}{3}$ and the point $(3, 2)$: $y - 2 = -\frac{5}{3}(x - 3).$

Step 3: Simplify to Slope-Intercept Form

Simplify the equation: $y - 2 = -\frac{5}{3}x + 5.$ $y = -\frac{5}{3}x + 7.$

Thus, the equation of the line is: $f(x) = -\frac{5}{3}x + 7.$

Step 4: Match with the Options

From the given choices, the correct answer is: A) $f(x) = -\frac{5}{3}x + 7$.

Let me know if you'd like a detailed explanation of any specific step!

Follow-up Questions:

1. What is the significance of the slope in the context of a linear function?
2. How would the equation change if $f(3) = 5$ and $f(6) = -1$?
3. Can you determine $f(x)$ if $f(3) = 2$ but no second point is given?
4. How do we interpret the $y$-intercept of the function $f(x) = -\frac{5}{3}x + 7$?
5. What changes to the equation occur if the slope becomes positive?

Tip:

When solving problems involving linear functions, always start by calculating the slope using two points. This sets the foundation for determining the equation in either point-slope or slope-intercept form!