Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
