Question

What are the domain and range of the function defined by y = |x + 3| - 2, written in set-builder notation and interval notation?

a) Domain: x ∈ ℝ, Range: y
b) Domain: x , Range: y ≥ -2
c) Domain: x ∈ ℝ, Range: y ≥ 2
d) Domain: x ≥ -3, Range: y

Answer 1

The function y = |x + 3| - 2 has a domain of all real numbers and a range of all real numbers greater than or equal to -2, both represented in set-builder and interval notation.

Step-by-step explanation:

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values) the function can take. For the function y = |x + 3| - 2, the absolute value expression inside the function implies that any real number can be an input, as all real numbers will yield a non-negative result after the absolute value operation. Therefore, the domain in set-builder notation is x , which corresponds to all real numbers, and in interval notation, it is (-∞, ∞). Looking at the range, since the absolute value ensures that the result is always non-negative, the smallest value y can take is when the inside of the absolute value is zero, which happens when x = -3. Subsequently, y would become -2 at that point. So, after the subtraction of 2, the smallest value of y is -2, which implies that y can be any real number greater than or equal to -2. The range in set-builder notation is y ≥ -2 and in interval notation, it is [-2, ∞).