To find the expression for the function $f(x)$, we need to determine the equation of the line passing through the points $(-6, -9)$ and $(-2, -1)$. The general form of a linear equation is:

$f(x) = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Step 1: Find the slope ($m$)

The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the given points $(-6, -9)$ and $(-2, -1)$:

$m = \frac{-1 - (-9)}{-2 - (-6)} = \frac{-1 + 9}{-2 + 6} = \frac{8}{4} = 2$

Step 2: Find the y-intercept ($b$)

Now that we know the slope $m = 2$, we can use one of the given points to find the y-intercept. Let's use the point $(-2, -1)$ and substitute into the equation $f(x) = 2x + b$.

$-1 = 2(-2) + b$

$-1 = -4 + b$

Solving for $b$:

$b = -1 + 4 = 3$

Step 3: Write the equation

Now that we have the slope $m = 2$ and the y-intercept $b = 3$, the equation of the line is:

$f(x) = 2x + 3$

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Would you like to explore any specific steps in more detail?

Here are some related questions:

1. How do you calculate the slope between any two points?
2. How do you find the y-intercept when given a point and the slope?
3. Can this method be applied to non-linear functions?
4. How would the equation change if the slope was negative?
5. What are the properties of lines parallel or perpendicular to this one?

Tip: Always double-check your slope calculation when dealing with negative numbers to avoid simple mistakes.