To determine the domain and range of the function [tex]\( y = (x+3)^2 - 5 \)[/tex], let's analyze the function step-by-step.
### Domain:
The domain of a function refers to the set of all possible input values (x-values) that the function can accept without any restrictions. Here, [tex]\( y = (x+3)^2 - 5 \)[/tex] is a quadratic function, which is defined for all real numbers.
So, the domain of [tex]\( y = (x+3)^2 - 5 \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range:
The range of a function refers to the set of all possible output values (y-values) that the function can produce.
To find the range of [tex]\( y = (x+3)^2 - 5 \)[/tex]:
1. Note that [tex]\((x+3)^2\)[/tex] is a quadratic function that opens upwards, meaning it has a minimum value at its vertex.
2. The vertex of [tex]\( (x+3)^2 \)[/tex] is at [tex]\( x = -3 \)[/tex], and the value at the vertex is 0; thus, when [tex]\( x = -3 \)[/tex], [tex]\( (x+3)^2 = 0 \)[/tex].
3. Plugging [tex]\( x = -3 \)[/tex] into the function gives us [tex]\( y = 0 - 5 = -5 \)[/tex].
Therefore, the minimum value of [tex]\( y = (x+3)^2 - 5 \)[/tex] is [tex]\(-5\)[/tex]. Since the parabola opens upwards, the range starts at [tex]\(-5\)[/tex] and extends to infinity.
So, the range of [tex]\( y = (x+3)^2 - 5 \)[/tex] is:
[tex]\[ [-5, \infty) \][/tex]
### Correct Answer:
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\([-5, \infty)\)[/tex]
From the given options, the correct choice is:
B.
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
[tex]\[ \text{Range: } [-5, \infty) \][/tex]
