Let's analyze the quadratic function [tex]\( y = (x + 3)^2 - 5 \)[/tex] to determine its domain and range.
Step-by-step solution:
1. Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For any quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex], the function is defined for all real numbers.
Thus, the domain of [tex]\( y = (x + 3)^2 - 5 \)[/tex] is all real numbers:
[tex]\( (-\infty, \infty) \)[/tex].
2. Range:
To determine the range, we need to understand the behavior and shape of the quadratic function. For a quadratic function in the form [tex]\( y = a(x - h)^2 + k \)[/tex], the vertex [tex]\((h, k)\)[/tex] represents the minimum or maximum point of the parabola.
For the given function [tex]\( y = (x + 3)^2 - 5 \)[/tex]:
- The function is in the vertex form [tex]\( y = a(x - h)^2 + k \)[/tex].
- Here, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-3, -5)\)[/tex].
- Since the coefficient of the squared term [tex]\((x + 3)^2\)[/tex] is positive, the parabola opens upwards.
This means the minimum value of the function occurs at the vertex [tex]\((-3, -5)\)[/tex]. The y-coordinate of the vertex is [tex]\(-5\)[/tex], which is the lowest value the function can take. As the parabola opens upwards, the function's values can go to infinity.
Thus, the range of [tex]\( y = (x + 3)^2 - 5 \)[/tex] is:
[tex]\( [-5, \infty) \)[/tex].
Given these detailed analyses:
- The domain is [tex]\( (-\infty, \infty) \)[/tex].
- The range is [tex]\( [-5, \infty) \)[/tex].
The correct answer is:
D. Domain: [tex]\( (-\infty, \infty) \)[/tex]
Range: [tex]\([ -5, \infty )\)[/tex]
