From the values provided, the function $f(x)$ demonstrates a pattern where specific values of $x$ correspond to given outputs $f(x)$. To identify the function, we analyze the data:

Given:

$f(-2) = -6,\quad f(-1) = -4,\quad f(0) = 0,\quad f(1) = 4,\quad f(2) = 6$

Observations:

• The function appears to be linear in nature because the changes in $f(x)$ are consistent for each step in $x$.
• Calculate the slope $m$: $m = \frac{\Delta f(x)}{\Delta x} = \frac{-4 - (-6)}{-1 - (-2)} = \frac{2}{1} = 2$ Thus, the function increases by 2 units in $f(x)$ for every 1 unit in $x$.

Linear Function Form:

A linear function is represented as: $f(x) = mx + c$ Where $m$ is the slope, and $c$ is the y-intercept.

Substitute Known Values:

Using $f(0) = 0$, we find $c$: $f(0) = 2(0) + c \implies c = 0$

Final Function:

The function is: $f(x) = 2x$

Verification:

• $f(-2) = 2(-2) = -6$ ✅
• $f(-1) = 2(-1) = -4$ ✅
• $f(0) = 2(0) = 0$ ✅
• $f(1) = 2(1) = 4$ ✅
• $f(2) = 2(2) = 6$ ✅

This function matches all the given points.

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Relative Questions:

1. How can we derive functions for non-linear patterns?
2. What are the steps to identify if a function is quadratic instead of linear?
3. What role does the slope play in determining the rate of change in functions?
4. How can the y-intercept affect the shape of a linear function?
5. Can a similar analysis be performed for higher-dimensional functions?

Tip:

When verifying a function, always test all given data points to ensure it matches perfectly.