To determine the domain and range of the function [tex]\(y = (x + 3)^2 - 5\)[/tex], we need to consider the behavior of the function and the values [tex]\(x\)[/tex] can take and the subsequent values [tex]\(y\)[/tex] can achieve.
Step-by-Step Solution:
1. Domain:
The domain of a function is the set of all possible values of [tex]\(x\)[/tex] for which the function is defined.
- The function [tex]\(y = (x + 3)^2 - 5\)[/tex] is a quadratic function.
- Quadratic functions are defined for all real numbers [tex]\(x\)[/tex].
- Therefore, the domain of [tex]\(y = (x + 3)^2 - 5\)[/tex] is all real numbers.
So, the domain is [tex]\((-\infty, \infty)\)[/tex].
2. Range:
The range of a function is the set of all possible values of [tex]\(y\)[/tex] that the function can produce.
- The expression [tex]\((x + 3)^2\)[/tex] is a perfect square and is always non-negative, meaning [tex]\((x + 3)^2 \geq 0\)[/tex].
- The minimum value of [tex]\((x + 3)^2\)[/tex] is [tex]\(0\)[/tex], which occurs when [tex]\(x = -3\)[/tex].
- Substituting [tex]\(x = -3\)[/tex] into the function gives [tex]\(y = (x + 3)^2 - 5 = 0 - 5 = -5\)[/tex].
- As [tex]\((x + 3)^2\)[/tex] contains all non-negative numbers, [tex]\((x + 3)^2 - 5\)[/tex] will produce all values starting from [tex]\(-5\)[/tex] and increasing to infinity.
Thus, the range of [tex]\(y = (x + 3)^2 - 5\)[/tex] is [tex]\([-5, \infty)\)[/tex].
Given these details, the correct answer is:
A. Domain: [tex]\((-\infty, \infty)\)[/tex]
Range: [tex]\([-5, \infty)\)[/tex]
