To determine which function has a domain of [tex]\( x \geq 5 \)[/tex] and a range of [tex]\( y \leq 3 \)[/tex], we need to analyze the domain and range for each provided function.
1. Function: [tex]\( y = \sqrt{x - 5} + 3 \)[/tex]
- Domain: The expression inside the square root [tex]\( x - 5 \)[/tex] must be non-negative. Therefore, [tex]\( x - 5 \geq 0 \)[/tex] implies [tex]\( x \geq 5 \)[/tex].
- Range: The square root function [tex]\( \sqrt{x - 5} \)[/tex] yields values starting from 0 and increasing upwards as [tex]\( x \)[/tex] increases. Therefore, [tex]\( y = \sqrt{x - 5} + 3 \)[/tex] starts at 3 and increases without bound. Thus, the range is [tex]\( y \geq 3 \)[/tex].
2. Function: [tex]\( y = \sqrt{x + 5} - 3 \)[/tex]
- Domain: The expression inside the square root [tex]\( x + 5 \)[/tex] must be non-negative. Therefore, [tex]\( x + 5 \geq 0 \)[/tex] implies [tex]\( x \geq -5 \)[/tex].
- Range: The square root function [tex]\( \sqrt{x + 5} \)[/tex] starts at 0 when [tex]\( x = -5 \)[/tex] and increases as [tex]\( x \)[/tex] increases. Therefore, [tex]\( y = \sqrt{x + 5} - 3 \)[/tex] starts at -3 and increases. Thus, the range is [tex]\( y \geq -3 \)[/tex].
3. Function: [tex]\( y = -\sqrt{x - 5} + 3 \)[/tex]
- Domain: The expression inside the square root [tex]\( x - 5 \)[/tex] must be non-negative. Therefore, [tex]\( x - 5 \geq 0 \)[/tex] implies [tex]\( x \geq 5 \)[/tex].
- Range: The negative square root function [tex]\( -\sqrt{x - 5} \)[/tex] starts at 0 when [tex]\( x = 5 \)[/tex] and decreases as [tex]\( x \)[/tex] increases. Therefore, [tex]\( y = -\sqrt{x - 5} + 3 \)[/tex] starts at 3 and decreases without bound. Thus, the range is [tex]\( y \leq 3 \)[/tex].
4. Function: [tex]\( y = -\sqrt{x + 5} - 3 \)[/tex]
- Domain: The expression inside the square root [tex]\( x + 5 \)[/tex] must be non-negative. Therefore, [tex]\( x + 5 \geq 0 \)[/tex] implies [tex]\( x \geq -5 \)[/tex].
- Range: The negative square root function [tex]\( -\sqrt{x + 5} \)[/tex] starts at 0 when [tex]\( x = -5 \)[/tex] and decreases as [tex]\( x \)[/tex] increases. Therefore, [tex]\( y = -\sqrt{x + 5} - 3 \)[/tex] starts at -3 and decreases. Thus, the range is [tex]\( y \leq -3 \)[/tex].
After analyzing each function, we can see that the function which has a domain of [tex]\( x \geq 5 \)[/tex] and a range of [tex]\( y \leq 3 \)[/tex] is:
[tex]\[ y = -\sqrt{x - 5} + 3 \][/tex]
Therefore, the correct answer is the third function:
[tex]\[ \boxed{3} \][/tex]
