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Determine the average value of the piecewise function f defined below on the interval x=-3 to x=6. Express your answer in simplest form. f(x)={[-3," for ",x < -2],[6," for ",x >= -2]:}

Determine the average value of the piecewise function f f defined below on the interval x=−3 x=-3 to x=6 x=6 . Express your answer in simplest form.\newlinef(x)={−3 for x<−26 for x≥−2 f(x)=\left\{\begin{array}{lll} -3 & \text { for } & x<-2 \\ 6 & \text { for } & x \geq-2 \end{array}\right.

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Q. Determine the average value of the piecewise function f f defined below on the interval x=−3 x=-3 to x=6 x=6 . Express your answer in simplest form.\newlinef(x)={−3 for x<−26 for x≥−2 f(x)=\left\{\begin{array}{lll} -3 & \text { for } & x<-2 \\ 6 & \text { for } & x \geq-2 \end{array}\right.
  1. Calculate Area - Part 11: For x<−2x < -2, f(x)=−3f(x) = -3. This applies from x=−3x = -3 to x=−2x = -2. We calculate the area under the curve for this part by multiplying the function value with the length of the interval. Area = f(x)×(upper bound−lower bound)=−3×(−2−(−3))=−3×1=−3f(x) \times (\text{upper bound} - \text{lower bound}) = -3 \times (-2 - (-3)) = -3 \times 1 = -3.
  2. Calculate Area - Part 22: For x≥−2x \geq -2, f(x)=6f(x) = 6. This applies from x=−2x = -2 to x=6x = 6. Again, we calculate the area under the curve for this part. Area = f(x)×(upper bound−lower bound)=6×(6−(−2))=6×8=48f(x) \times (\text{upper bound} - \text{lower bound}) = 6 \times (6 - (-2)) = 6 \times 8 = 48.
  3. Add Total Areas: Now, we add the areas of both parts to get the total area under the curve.\newlineTotal area = area from x=−3x=-3 to x=−2x=-2 + area from x=−2x=-2 to x=6x=6 = −3+48=45-3 + 48 = 45.
  4. Find Average Value: To find the average value, we divide the total area by the length of the entire interval.\newlineLength of the interval = upper bound - lower bound = 6−(−3)=96 - (-3) = 9.\newlineAverage value = Total area / Length of the interval = 459=5\frac{45}{9} = 5.

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