To find the initial value \( b \) of the linear function, we can use the formula for a linear function in slope-intercept form, which is:
\[ y = mx + b \]
where \( m \) is the slope (rate of change) and \( b \) is the y-intercept (initial value).
Given:
- The rate of change \( m = -47 \)
- The point \( (x, y) = (14, 3) \)
We can substitute the values into the equation:
\[ 3 = -47(14) + b \]
Calculating \( -47(14) \):
\[ -47 \times 14 = -658 \]
Now substituting back in:
\[ 3 = -658 + b \]
To solve for \( b \), add \( 658 \) to both sides:
\[ b = 3 + 658 \]
\[ b = 661 \]
None of the provided responses match this result, indicating that there may have been a misunderstanding or mistake in the provided options.
The correct calculation determines that the initial value \( b \) is \( 661 \). If you meant to look for a closer set of calculations or assumptions, please clarify!