What are the domain and range of the function [tex]f(x)=-\sqrt{x+3}-2[/tex]?

A. Domain: [tex]-3 \leq x \leq -2[/tex], Range: [tex]y \geq -2[/tex]

B. Domain: [tex]-3 \leq x \leq -2[/tex], Range: [tex]y \leq -2[/tex]

C. Domain: [tex]x \geq -3[/tex], Range: [tex]y \leq -2[/tex]

D. Domain: [tex]x \geq -3[/tex], Range: [tex]y \geq -2[/tex]



Answer :

Let's determine the domain and range of the function [tex]\( f(x) = -\sqrt{x + 3} - 2 \)[/tex].

### Domain
The domain of a function defines the set of all possible input values (x-values) for which the function is defined.

1. Identify the restriction:
The function contains a square root, [tex]\( \sqrt{x + 3} \)[/tex]. The expression inside the square root, [tex]\( x + 3 \)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers.

2. Set the restriction:
Therefore, we must have:
[tex]\[ x + 3 \geq 0 \][/tex]

3. Solve the inequality:
[tex]\[ x + 3 \geq 0 \implies x \geq -3 \][/tex]

Hence, the domain of the function [tex]\( f(x) = -\sqrt{x + 3} - 2 \)[/tex] is:
[tex]\[ \boxed{x \geq -3} \][/tex]

### Range
The range of a function defines the set of all possible output values (y-values).

1. Analyze the function's behavior:
The function [tex]\( f(x) = -\sqrt{x + 3} - 2 \)[/tex] is the negative of the square root function, shifted downward by 2 units.

2. Evaluate [tex]\( f(x) \)[/tex] at the boundary of the domain:
At the leftmost boundary of the domain ([tex]\( x = -3 \)[/tex]):
[tex]\[ f(-3) = -\sqrt{-3 + 3} - 2 = -\sqrt{0} - 2 = -0 - 2 = -2. \][/tex]

3. Consider the behavior as [tex]\( x \)[/tex] increases:
As [tex]\( x \)[/tex] increases from [tex]\( -3 \)[/tex] toward positive infinity, [tex]\( \sqrt{x + 3} \)[/tex] will increase. Since there is a negative sign in front of the square root, [tex]\( -\sqrt{x + 3} \)[/tex] will decrease, making the values more negative.

Therefore, as [tex]\( x \)[/tex] increases:
[tex]\[ f(x) = -\sqrt{x + 3} - 2 \leq -2 \][/tex]

Thus, the range of function [tex]\( f(x) = -\sqrt{x + 3} - 2 \)[/tex] is:
[tex]\[ \boxed{y \leq -2} \][/tex]

Given the analysis above, the correct answers for the domain and range of the function [tex]\( f(x) = -\sqrt{x + 3} - 2 \)[/tex] are:
[tex]\[ \boxed{\text{domain: } x \geq -3, \text{ range: } y \leq -2} \][/tex]