To determine which graph corresponds to the function [tex]\( y = \sqrt{-x-3} \)[/tex], we need to analyze the properties and constraints of the function.
1. Domain Analysis:
- The expression inside the square root, [tex]\(-x-3\)[/tex], must be non-negative for the square root to be real and defined.
- Therefore, [tex]\(-x - 3 \geq 0\)[/tex].
- Solving this inequality:
[tex]\[
-x - 3 \geq 0 \implies -x \geq 3 \implies x \leq -3
\][/tex]
- The domain of the function is [tex]\( x \leq -3 \)[/tex].
2. Range Analysis:
- Since the square root function outputs non-negative values, [tex]\( y \)[/tex] is always greater than or equal to 0.
- Therefore, the range of the function is [tex]\( y \geq 0 \)[/tex].
3. Behavior and Shape:
- As [tex]\( x \)[/tex] approaches [tex]\(-3\)[/tex] from the left, we can check the behavior of [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt{-x-3} \implies \text{if } x = -3, \text{ then } y = \sqrt{-(-3)-3} = \sqrt{0} = 0
\][/tex]
- For values [tex]\( x < -3 \)[/tex], [tex]\(-x-3\)[/tex] becomes positive, and thus [tex]\( \sqrt{-x-3} \)[/tex] is defined and positive.
- As [tex]\( x \)[/tex] decreases further from [tex]\(-3\)[/tex] to more negative values, [tex]\(-x-3\)[/tex] increases and the value of [tex]\( y \)[/tex] increases.
4. Graph Characteristics:
- The graph will start at [tex]\((x,y) = (-3, 0)\)[/tex] and will extend to the left with increasing values of [tex]\( y \)[/tex].
- This is part of a parabolic shape translated horizontally.
Considering these points, the correct graph for [tex]\( y = \sqrt{-x-3} \)[/tex] is a curve that:
- Starts at the point [tex]\((-3, 0)\)[/tex].
- Extends to the left for [tex]\( x \leq -3 \)[/tex].
- Rises upward as [tex]\( x \)[/tex] becomes more negative.
This ensures the graph correctly represents the domain [tex]\( x \leq -3 \)[/tex] and the range [tex]\( y \geq 0 \)[/tex]. Make sure to select the graph that fits these observed characteristics.
