We are tasked with finding a linear function $f(x)$ that passes through the given points. The points provided are:

$(-4, -2), (-2, 0), (0, -2), (-1, 0)$

Since a linear function is of the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept, we can use two points to find the slope $m$.

Step 1: Find the slope $m$

To calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the formula:

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

Let's take two points from the list, say $(-4, -2)$ and $(-2, 0)$:

$m = \frac{0 - (-2)}{-2 - (-4)} = \frac{2}{2} = 1$

So, the slope of the line is $m = 1$.

Step 2: Find the y-intercept $b$

Now that we know the slope is $m = 1$, we can use one of the points to find the y-intercept $b$. Using the point $(-2, 0)$:

$f(x) = mx + b$ $0 = 1(-2) + b$ $0 = -2 + b$ $b = 2$

Step 3: Write the function

Now we have the slope $m = 1$ and the y-intercept $b = 2$, so the linear function is:

$f(x) = x + 2$

Verification

We can verify this by plugging in the given x-values:

• For $x = -4$, $f(-4) = -4 + 2 = -2$.
• For $x = -2$, $f(-2) = -2 + 2 = 0$.
• For $x = 0$, $f(0) = 0 + 2 = 2$ (but your table lists $-2$, which may indicate a mistake in the data provided).

Thus, the function $f(x) = x + 2$ fits most of the points. There may be a data inconsistency in the question.

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Would you like me to go into more detail or clarify anything?

Here are some related questions:

1. How can we derive the equation of a linear function from two points?
2. What happens if the data for a linear function doesn't follow the expected pattern?
3. How do you check whether a function fits all points of a given table?
4. How do you find the slope and y-intercept using a set of points?
5. How would you approach finding a linear function with more than two points?

Tip: Always double-check the data for consistency before deriving the equation, especially if the function doesn't match all points.