To determine the value of [tex]\( f(3) \)[/tex] for the linear function given by the points in the table, we need to follow these steps:
1. Identify two points to calculate the slope:
We can select two points from the given table to determine the slope [tex]\( m \)[/tex] of the linear function. Let's use the points [tex]\( (0, -2) \)[/tex] and [tex]\( (2, 4) \)[/tex].
2. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] of a linear function is calculated using the formula:
[tex]\[
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Substituting the chosen points [tex]\( (x_1, f(x_1)) = (0, -2) \)[/tex] and [tex]\( (x_2, f(x_2)) = (2, 4) \)[/tex], we get:
[tex]\[
m = \frac{4 - (-2)}{2 - 0} = \frac{4 + 2}{2} = \frac{6}{2} = 3
\][/tex]
3. Determine the y-intercept [tex]\( b \)[/tex]:
The slope-intercept form of a linear function is [tex]\( f(x) = mx + b \)[/tex]. We already have [tex]\( m = 3 \)[/tex]. To find [tex]\( b \)[/tex], we use one of the points, say [tex]\( (0, -2) \)[/tex], and the equation:
[tex]\[
f(x) = 3x + b
\][/tex]
Plugging [tex]\( x = 0 \)[/tex] and [tex]\( f(x) = -2 \)[/tex]:
[tex]\[
-2 = 3(0) + b \implies b = -2
\][/tex]
4. Formulate the linear function:
Substituting [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the general form, the linear function becomes:
[tex]\[
f(x) = 3x + -2 \implies f(x) = 3x - 2
\][/tex]
5. Calculate [tex]\( f(3) \)[/tex]:
Using the linear function [tex]\( f(x) = 3x - 2 \)[/tex], we find [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = 3(3) - 2 = 9 - 2 = 7
\][/tex]
Hence, the value of [tex]\( f(3) \)[/tex] is [tex]\(\boxed{7}\)[/tex].
