Questions: f(x) is defined as follows: - For 0 < x < 3, f(x) = -(x-3)^2 + 8 - For x = 3, f(x) = 3 - For 3 < x < 6, f(x) = -x + 7 Find f(1).

Transcript text: \[ f(x)=\left\{\begin{array}{lll} -(x-3)^{2}+8 & \text { for } & 0

Solution

Solution Steps

To find \( f(1) \), we need to determine which piece of the piecewise function applies to \( x = 1 \). Since \( 0 < 1 < 3 \), we use the first piece of the function, which is \( f(x) = -(x-3)^2 + 8 \). Substitute \( x = 1 \) into this expression to find \( f(1) \).

Step 1: Identify the Relevant Piece of the Function

The function \( f(x) \) is defined piecewise. To find \( f(1) \), we need to determine which piece of the function applies to \( x = 1 \). The condition \( 0 < x < 3 \) applies, so we use the expression \( f(x) = -(x-3)^2 + 8 \).

Step 2: Substitute \( x = 1 \) into the Expression

Substitute \( x = 1 \) into the expression \( f(x) = -(x-3)^2 + 8 \): \[ f(1) = -((1) - 3)^2 + 8 \]

Step 3: Simplify the Expression

Calculate the value: \[ f(1) = -(-2)^2 + 8 = -4 + 8 = 4 \]