To determine the domain and range of the function [tex]\( f(x) = -(x+5)^2 - 4 \)[/tex], let's analyze the function step by step.
### Domain:
The domain of a function is the set of all possible input values (x-values) that the function can accept.
For the function [tex]\( f(x) = -(x+5)^2 - 4 \)[/tex]:
- This is a quadratic function (a parabola), and quadratic functions are defined for all real numbers.
- There are no restrictions on the x-values for which this function is defined.
Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers, which can be expressed in interval notation as:
[tex]\[
(-\infty, \infty)
\][/tex]
### Range:
The range of a function is the set of all possible output values (y-values) that the function can produce.
For the function [tex]\( f(x) = -(x+5)^2 - 4 \)[/tex]:
- This is a downward-facing parabola because the coefficient of [tex]\( (x+5)^2 \)[/tex] is negative.
- The vertex of the parabola, which is the maximum point, occurs at [tex]\( (h, k) \)[/tex], where [tex]\( h = -5 \)[/tex] and [tex]\( k = -4 \)[/tex].
- The maximum value of [tex]\( f(x) \)[/tex] is [tex]\( -4 \)[/tex], occurring when [tex]\( x = -5 \)[/tex].
- Since the parabola opens downwards, the function values will be less than or equal to this maximum point.
Therefore, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[
(-\infty, -4]
\][/tex]
Putting everything together, the domain and range of the function are:
Domain: [tex]\((-\infty, \infty)\)[/tex]
Range: [tex]\((-\infty, -4]\)[/tex]
So, we fill in the given interval notations accordingly:
Domain: [tex]\((-\infty, \infty)\)[/tex] and Range: [tex]\((-\infty, -4]\)[/tex]
