Let's break down the function [tex]\( y = (x + 3)^2 - 5 \)[/tex] to understand its domain and range:
### Domain
The domain of a function refers to all the possible values of [tex]\(x\)[/tex] that can be input into the function.
In the function [tex]\( y = (x + 3)^2 - 5 \)[/tex], there are no restrictions on [tex]\(x\)[/tex]. This is because the term [tex]\((x + 3)^2\)[/tex] is a polynomial expression, which can handle any real number input. Therefore, [tex]\(x\)[/tex] can be any real number.
So, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range
The range of a function refers to all the possible values that [tex]\(y\)[/tex] can take as [tex]\(x\)[/tex] varies over the domain.
The function [tex]\( y = (x + 3)^2 - 5 \)[/tex] is in the form of a quadratic function, which represents a parabola. The standard form of a quadratic function is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola and the parabola opens upwards if [tex]\(a > 0\)[/tex] and downwards if [tex]\(a < 0\)[/tex].
For [tex]\( y = (x + 3)^2 - 5 \)[/tex]:
- The parabola opens upwards because the coefficient of the squared term [tex]\((x + 3)^2\)[/tex] is positive (1).
- The vertex of the parabola can be obtained directly from the function. Here, [tex]\( (h, k) \)[/tex] is [tex]\((-3, -5)\)[/tex].
As the parabola opens upwards, the smallest value of [tex]\(y\)[/tex] is at the vertex, which corresponds to [tex]\( y = -5 \)[/tex]. Therefore, the minimum value of [tex]\( y \)[/tex] is [tex]\(-5\)[/tex], and as [tex]\(x\)[/tex] moves away from the vertex in either direction, [tex]\( y \)[/tex] increases without bound.
Thus, the range of the function is:
[tex]\[ [-5, \infty) \][/tex]
Matching our findings with the provided options:
- The domain is [tex]\( (-\infty, \infty) \)[/tex].
- The range is [tex]\( [-5, \infty) \)[/tex].
Therefore, the correct answer is:
D. Domain: [tex]\( (-\infty, \infty) \)[/tex]
Range: [tex]\([-5, \infty)\)[/tex]
