To find the equation that represents the function in the table, we can observe the relationship between the values of \( x \) and \( y \).
Here's the data provided:
\[ \begin{align*} x & : 0, 1, 2, 3 \ y & : -5, -9, -13, -17 \ \end{align*} \]
Calculating the changes in \( y \):
- From \( x = 0 \) to \( x = 1 \): \( -9 - (-5) = -4 \)
- From \( x = 1 \) to \( x = 2 \): \( -13 - (-9) = -4 \)
- From \( x = 2 \) to \( x = 3 \): \( -17 - (-13) = -4 \)
The change in \( y \) is consistent at \(-4\) for each increase in \( x \) by 1, indicating that the slope (\( m \)) of the linear function is \(-4\).
Now we can determine the equation of the line in the form \( y = mx + b \). We can use one of the points to solve for \( b \). Choosing the point \( (0, -5) \) gives:
\[ -5 = -4(0) + b \implies b = -5 \]
Substituting \( m \) and \( b \) into the equation gives:
\[ y = -4x - 5 \]
In function notation, this would be:
\[ f(x) = -4x - 5 \]
Thus, the correct equation that describes the function in the table using function notation is:
f(x) = -4x - 5.