We are given a table of values for a function $f(x)$ and we want to estimate the value of $f(2)$. The table provides the following data: $x = -5$, $f(x) = -2$ $x = 3$, $f(x) = 2$ $x = 5$, $f(x) = 3$

1. Problem Description

We are given a table of values for a function $f(x)$ and we want to estimate the value of $f(2)$. The table provides the following data:

$x = -5$, $f(x) = -2$

$x = 3$, $f(x) = 2$

$x = 5$, $f(x) = 3$

2. Solution Steps

We can use linear interpolation to estimate the value of $f(2)$. Since $x=2$ is between $x=-5$ and $x=3$, we can consider the points $(-5, -2)$ and $(3, 2)$.

The equation of the line passing through the points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$

Using the points $(-5, -2)$ and $(3, 2)$, we have:

$y - (-2) = \frac{2 - (-2)}{3 - (-5)} (x - (-5))$

$y + 2 = \frac{4}{8} (x + 5)$

$y + 2 = \frac{1}{2} (x + 5)$

$y = \frac{1}{2}x + \frac{5}{2} - 2$

$y = \frac{1}{2}x + \frac{5}{2} - \frac{4}{2}$

$y = \frac{1}{2}x + \frac{1}{2}$

Now, we can estimate $f(2)$ by plugging in $x = 2$:

$f(2) = \frac{1}{2}(2) + \frac{1}{2}$

$f(2) = 1 + \frac{1}{2}$

$f(2) = \frac{3}{2}$

3. Final Answer

$f(2) = \frac{3}{2}$