Final answer:
After calculating the slope of the linear function as 5 and determining the y-intercept as 13, it was found that none of the provided options match the linear function that satisfies the conditions f(-3) = -2 and f(-2) = 3. This indicates a possible mistake in the given options.
Step-by-step explanation:
To write a linear function that satisfies the given conditions — f(-3) = -2 and f(-2) = 3 — we need to determine the function's slope and y-intercept. First, we calculate the slope (m) using the two given points:
Slope (m) =
\(
\frac{f(-2) - f(-3)}{-2 - (-3)}
\) =
\(
\frac{3 - (-2)}{1}
\) =
\(
\frac{5}{1}
\) = 5
Thus, the slope of the line is 5. Next, using the slope-intercept form of a line, y = mx + b, we substitute one of the given points (-3, -2) into the equation:
-2 = 5(-3) + b
-2 = -15 + b
b = 13
Now we have both the slope and the y-intercept, so the linear function is f(x) = 5x + 13. However, among the given choices, there is no direct match. Let's test the options:
- For option A, f(-3) = -2(-3) + 3 = 6 + 3 = 9 (does not satisfy f(-3) = -2)
- For option B, f(-3) = 2(-3) - 3 = -6 - 3 = -9 (does not satisfy f(-3) = -2)
- For option C, f(-3) = -5(-3) + 1 = 15 + 1 = 16 (does not satisfy f(-3) = -2)
- For option D, f(-3) = (-3) + 2 = -1 (does not satisfy f(-3) = -2)
None of the options matches the linear function with the given conditions, suggesting a possible error in the options provided or in the calculation.