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{:[f(x)={[-(x+1)^(2)+3," for ",x <= -1],[-2x+2," for ",-1 < x < 3],[-4," for ",x >= 3]:}],[" Find "f(3)]:} Answer:

f(x)={−(x+1)2+3 for x≤−1−2x+2 for −1<x<3−4 for x≥3 Find f(3) \begin{array}{l} f(x)=\left\{\begin{array}{lll} -(x+1)^{2}+3 & \text { for } & x \leq-1 \\ -2 x+2 & \text { for } & -1<x<3 \\ -4 & \text { for } & x \geq 3 \end{array}\right. \\ \text { Find } f(3) \\ \end{array} \newlineAnswer:\newline

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Q. f(x)={−(x+1)2+3 for x≤−1−2x+2 for −1<x<3−4 for x≥3 Find f(3) \begin{array}{l} f(x)=\left\{\begin{array}{lll} -(x+1)^{2}+3 & \text { for } & x \leq-1 \\ -2 x+2 & \text { for } & -1<x<3 \\ -4 & \text { for } & x \geq 3 \end{array}\right. \\ \text { Find } f(3) \\ \end{array} \newlineAnswer:\newline
  1. Identify Piece for x=3x = 3: Identify which piece of the piecewise function applies to x=3x = 3. We have three pieces to consider: 11. f(x)=−(x+1)2+3f(x) = -(x+1)^2 + 3 for x≤−1x \leq -1 22. f(x)=−2x+2f(x) = -2x + 2 for −1<x<3-1 < x < 3 33. f(x)=−4f(x) = -4 for x≥3x \geq 3 Since we are looking for f(3)f(3), we use the third piece because 33 is greater than or equal to 33.
  2. Apply x=3x = 3: Apply the value x=3x = 3 to the appropriate piece of the function.\newlineUsing the third piece, we have:\newlinef(3)=−4f(3) = -4\newlineThere is no calculation needed here since the function value is a constant for x≥3x \geq 3.

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