1. The problem asks to determine if the function represented by the table is linear or nonlinear. 2. A function is linear if it can be expressed as $y = mx + b$, where $m$ and $b$ are constants, and the rate of change (slope) between any two points is constant. 3. Let's examine the given points from the table: Points: $(-8, 2)$, $(5, 8)$, $(2, 4)$, $(5, -6)$, $(8, 2)$, $(5, -10)$ 4. Calculate the slope between pairs of points where $x$ values differ: Between $(-8, 2)$ and $(5, 8)$: $$m = \frac{8 - 2}{5 - (-8)} = \frac{6}{13}$$ Between $(5, 8)$ and $(2, 4)$: $$m = \frac{4 - 8}{2 - 5} = \frac{-4}{-3} = \frac{4}{3}$$ 5. Since the slopes $\frac{6}{13}$ and $\frac{4}{3}$ are not equal, the rate of change is not constant. 6. Additionally, the $x$ value 5 corresponds to multiple different $y$ values (8, -6, -10), which violates the definition of a function. 7. Therefore, the function is nonlinear and not a valid function due to multiple $y$ values for the same $x$. Final answer: The function is nonlinear and not a valid function.