Answer :
To determine the graph of the function \( y = \sqrt{-x-3} \), we need to define the domain and compute corresponding \( y \)-values for given \( x \)-values.
### Step 1: Determine the Domain
The function \( y = \sqrt{-x-3} \) is defined only when the expression inside the square root is non-negative. Therefore, we need:
[tex]\[ -x - 3 \geq 0 \][/tex]
[tex]\[ -x \geq 3 \][/tex]
[tex]\[ x \leq -3 \][/tex]
Thus, the domain of the function is \( x \leq -3 \).
### Step 2: Compute Corresponding \( y \)-Values
Next, let's compute the \( y \)-values for a range of \( x \)-values within the domain. We'll take \( x \) in the region \( [-10, -3] \) and compute \( y \).
For each \( x \in [-10, -3] \):
1. \( x = -3 \)
[tex]\[ y = \sqrt{-(-3) - 3} = \sqrt{0} = 0 \][/tex]
2. \( x = -4 \)
[tex]\[ y = \sqrt{-(-4) - 3} = \sqrt{1} = 1 \][/tex]
3. \( x = -5 \)
[tex]\[ y = \sqrt{-(-5) - 3} = \sqrt{2} \approx 1.414 \][/tex]
4. \( x = -6 \)
[tex]\[ y = \sqrt{-(-6) - 3} = \sqrt{3} \approx 1.732 \][/tex]
5. \( x = -7 \)
[tex]\[ y = \sqrt{-(-7) - 3} = \sqrt{4} = 2 \][/tex]
6. \( x = -8 \)
[tex]\[ y = \sqrt{-(-8) - 3} = \sqrt{5} \approx 2.236 \][/tex]
7. \( x = -9 \)
[tex]\[ y = \sqrt{-(-9) - 3} = \sqrt{6} \approx 2.449 \][/tex]
8. \( x = -10 \)
[tex]\[ y = \sqrt{-(-10) - 3} = \sqrt{7} \approx 2.646 \][/tex]
### Step 3: List the Points
From the computations, we obtain the following points \( (x, y) \):
[tex]\[ \begin{aligned} & (-3, 0) \\ & (-4, 1) \\ & (-5, \sqrt{2}) \approx (-5, 1.414) \\ & (-6, \sqrt{3}) \approx (-6, 1.732) \\ & (-7, 2) \\ & (-8, \sqrt{5}) \approx (-8, 2.236) \\ & (-9, \sqrt{6}) \approx (-9, 2.449) \\ & (-10, \sqrt{7}) \approx (-10, 2.646) \\ \end{aligned} \][/tex]
### Step 4: Analyze the Graph
Plotting these points, we notice that as \( x \) becomes more negative (moving leftwards), \( y \) increases. The curve starts at \( (-3, 0) \) and increasingly bulges upwards towards more negative \( x \)-values.
The graph of \( y = \sqrt{-x-3} \) is a curve that starts from the point \((-3,0)\), moves leftwards and upwards, increasing gradually.
### Conclusion
The correct graph is one that meets these criteria: starting at \( (-3, 0) \) and increasing upwards as \( x \) values decrease. This should be evident visually by comparing the given graphs with these computations.
By these steps, it is clear which graph among the options is correct - the graph that starts from [tex]\((-3, 0)\)[/tex] and ascends gradually as [tex]\( x \)[/tex] becomes more negative. This graphical analysis leads to proper identification without further ambiguity.
### Step 1: Determine the Domain
The function \( y = \sqrt{-x-3} \) is defined only when the expression inside the square root is non-negative. Therefore, we need:
[tex]\[ -x - 3 \geq 0 \][/tex]
[tex]\[ -x \geq 3 \][/tex]
[tex]\[ x \leq -3 \][/tex]
Thus, the domain of the function is \( x \leq -3 \).
### Step 2: Compute Corresponding \( y \)-Values
Next, let's compute the \( y \)-values for a range of \( x \)-values within the domain. We'll take \( x \) in the region \( [-10, -3] \) and compute \( y \).
For each \( x \in [-10, -3] \):
1. \( x = -3 \)
[tex]\[ y = \sqrt{-(-3) - 3} = \sqrt{0} = 0 \][/tex]
2. \( x = -4 \)
[tex]\[ y = \sqrt{-(-4) - 3} = \sqrt{1} = 1 \][/tex]
3. \( x = -5 \)
[tex]\[ y = \sqrt{-(-5) - 3} = \sqrt{2} \approx 1.414 \][/tex]
4. \( x = -6 \)
[tex]\[ y = \sqrt{-(-6) - 3} = \sqrt{3} \approx 1.732 \][/tex]
5. \( x = -7 \)
[tex]\[ y = \sqrt{-(-7) - 3} = \sqrt{4} = 2 \][/tex]
6. \( x = -8 \)
[tex]\[ y = \sqrt{-(-8) - 3} = \sqrt{5} \approx 2.236 \][/tex]
7. \( x = -9 \)
[tex]\[ y = \sqrt{-(-9) - 3} = \sqrt{6} \approx 2.449 \][/tex]
8. \( x = -10 \)
[tex]\[ y = \sqrt{-(-10) - 3} = \sqrt{7} \approx 2.646 \][/tex]
### Step 3: List the Points
From the computations, we obtain the following points \( (x, y) \):
[tex]\[ \begin{aligned} & (-3, 0) \\ & (-4, 1) \\ & (-5, \sqrt{2}) \approx (-5, 1.414) \\ & (-6, \sqrt{3}) \approx (-6, 1.732) \\ & (-7, 2) \\ & (-8, \sqrt{5}) \approx (-8, 2.236) \\ & (-9, \sqrt{6}) \approx (-9, 2.449) \\ & (-10, \sqrt{7}) \approx (-10, 2.646) \\ \end{aligned} \][/tex]
### Step 4: Analyze the Graph
Plotting these points, we notice that as \( x \) becomes more negative (moving leftwards), \( y \) increases. The curve starts at \( (-3, 0) \) and increasingly bulges upwards towards more negative \( x \)-values.
The graph of \( y = \sqrt{-x-3} \) is a curve that starts from the point \((-3,0)\), moves leftwards and upwards, increasing gradually.
### Conclusion
The correct graph is one that meets these criteria: starting at \( (-3, 0) \) and increasing upwards as \( x \) values decrease. This should be evident visually by comparing the given graphs with these computations.
By these steps, it is clear which graph among the options is correct - the graph that starts from [tex]\((-3, 0)\)[/tex] and ascends gradually as [tex]\( x \)[/tex] becomes more negative. This graphical analysis leads to proper identification without further ambiguity.
