Final answer:
To find f(1) for the given piecewise function, we use the appropriate interval for x, which is -3 ≤ x ≤ 1. Substituting x = 1 into the expression for this interval, we get f(1) = 1 + 4, which simplifies to f(1) = 5.
Step-by-step explanation:
To find f(1), we should look at the definition of the function f(x) and determine which part of the piecewise function applies when x equals 1. The function definition provides three expressions for different intervals of x:
For x < -3: f(x) = -(2/3)x - 3
For -3 ≤ x ≤ 1: f(x) = x + 4
For x > 1: f(x) = -(x - 3)² - 2
Because we are looking for f(1), we use the second definition, which holds for -3 ≤ x ≤ 1. Hence, we substitute x with 1:
f(1) = 1 + 4
f(1) = 5
This calculation shows us that the value of the function f(x) at x = 1 is 5.