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Sagot :
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]
### 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]
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