Final answer:
The function f(x) = |x+3| - 1 is positive on the interval (-2, ∞) and negative on the interval (-∞, -3), with a point of zero at x = -2.
Step-by-step explanation:
We are tasked with identifying the intervals on which the function f(x) = |x+3| - 1 is positive and negative. To determine this, we need to analyze the behavior of the absolute value function.
Firstly, we recognize that the absolute value function, |x+3|, is equal to x+3 when x+3 ≥ 0 (or x ≥ -3) and equal to -(x+3) when x+3 < 0 (or x < -3). We can use these two cases to rewrite f(x) as:
- For x ≥ -3: f(x) = x+3 - 1 = x+2.
- For x < -3: f(x) = -(x+3) - 1 = -x-4.
Now, let's find the intervals where f(x) is positive or negative:
- For x ≥ -3, the function f(x) = x+2 is positive when x > -2. Hence, the interval is (-2, ∞).
- For x < -3, the function f(x) = -x-4 is always negative, since for any negative x, the subtraction of a larger absolute value number will remain negative. Thus, the interval is (-∞, -3).
Combining these results, we can conclude that the function f(x) is positive on the interval (-2, ∞) and negative on the interval (-∞, -3). There is also a point of zero at x = -2.