Question

f(x) = \

$$\begin{cases} -\\frac{2}{3}x - 3 & \\text{for } x < -3 \\\\ x + 4 & \\text{for } -3 \\leq x \\leq 1 \\\\ -(x - 3)^2 - 2 & \\text{for } x > 1 \\end{cases}$$

\\\\ \text{find } f(1)

Explanation:

Step1: Determine the applicable piece

We need to find \( f(1) \). Looking at the domain conditions: for \( x > 1 \), the function is \( (x - 3)^2 - 2 \); for \( -3\leq x\leq1 \), it's \( x + 4 \); for \( x < -3 \), it's \( -\frac{2}{3}x - 3 \). Since \( 1 \) is in the interval \( -3\leq x\leq1 \), we use the piece \( f(x)=x + 4 \).

Step2: Substitute \( x = 1 \) into the piece

Substitute \( x = 1 \) into \( f(x)=x + 4 \). So \( f(1)=1 + 4 \).

Step3: Calculate the result

\( 1+4 = 5 \).

Answer:

\( 5 \)