To find the equation of the linear function that models the relationship between \(x\) and \(y\) given in the table, we can first identify the slope (\(m\)) using the two points \((4, 38)\) and \((6, 58)\).
The formula for the slope is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from the points:
\[ m = \frac{58 - 38}{6 - 4} = \frac{20}{2} = 10 \]
Now that we know the slope (\(m = 10\)), we can use the point-slope form of the equation to find the equation of the line. We can use the point \( (4, 38) \):
The point-slope form of the equation is:
\[ y - y_1 = m(x - x_1) \]
Substituting in the slope and the point:
\[ y - 38 = 10(x - 4) \]
Distributing the slope on the right side:
\[ y - 38 = 10x - 40 \]
Now, add 38 to both sides to solve for \(y\):
\[ y = 10x - 40 + 38 \] \[ y = 10x - 2 \]
Thus, the equation of the linear function is:
\[ \boxed{y = 10x - 2} \]
This matches the response \(y = 10x - 2\).