Answer :
To determine the range of the function [tex]\( y = \sqrt{x+5} \)[/tex], we need to understand the behavior of the function and the values that [tex]\( y \)[/tex] can take.
1. Identify the domain of the function: Since we have a square root function, the expression inside the square root, [tex]\( x + 5 \)[/tex], must be non-negative. This gives us:
[tex]\[ x + 5 \geq 0 \implies x \geq -5 \][/tex]
So, the domain of the function is [tex]\( x \geq -5 \)[/tex].
2. Determine the range from the domain: For the given domain, we substitute the values of [tex]\( x \)[/tex] to see what values [tex]\( y \)[/tex] can take.
- When [tex]\( x = -5 \)[/tex]:
[tex]\[ y = \sqrt{-5 + 5} = \sqrt{0} = 0 \][/tex]
Hence, the smallest value that [tex]\( y \)[/tex] can take is 0.
- As [tex]\( x \)[/tex] increases (i.e., [tex]\( x > -5 \)[/tex]), the value of [tex]\( x + 5 \)[/tex] becomes positive, and the square root of any positive number is also positive:
[tex]\[ \text{For } x > -5, \quad y = \sqrt{x + 5} > 0 \][/tex]
3. Analyze the possible values of [tex]\( y \)[/tex]: Since [tex]\( y = \sqrt{x+5} \)[/tex] can only produce 0 when [tex]\( x = -5 \)[/tex] and a positive value when [tex]\( x > -5 \)[/tex], [tex]\( y \)[/tex] will take all values greater than or equal to 0.
Thus, the range of the function [tex]\( y = \sqrt{x + 5} \)[/tex] is:
[tex]\[ y \geq 0 \][/tex]
Therefore, the correct answer is:
[tex]\[ y \geq 0 \][/tex]
1. Identify the domain of the function: Since we have a square root function, the expression inside the square root, [tex]\( x + 5 \)[/tex], must be non-negative. This gives us:
[tex]\[ x + 5 \geq 0 \implies x \geq -5 \][/tex]
So, the domain of the function is [tex]\( x \geq -5 \)[/tex].
2. Determine the range from the domain: For the given domain, we substitute the values of [tex]\( x \)[/tex] to see what values [tex]\( y \)[/tex] can take.
- When [tex]\( x = -5 \)[/tex]:
[tex]\[ y = \sqrt{-5 + 5} = \sqrt{0} = 0 \][/tex]
Hence, the smallest value that [tex]\( y \)[/tex] can take is 0.
- As [tex]\( x \)[/tex] increases (i.e., [tex]\( x > -5 \)[/tex]), the value of [tex]\( x + 5 \)[/tex] becomes positive, and the square root of any positive number is also positive:
[tex]\[ \text{For } x > -5, \quad y = \sqrt{x + 5} > 0 \][/tex]
3. Analyze the possible values of [tex]\( y \)[/tex]: Since [tex]\( y = \sqrt{x+5} \)[/tex] can only produce 0 when [tex]\( x = -5 \)[/tex] and a positive value when [tex]\( x > -5 \)[/tex], [tex]\( y \)[/tex] will take all values greater than or equal to 0.
Thus, the range of the function [tex]\( y = \sqrt{x + 5} \)[/tex] is:
[tex]\[ y \geq 0 \][/tex]
Therefore, the correct answer is:
[tex]\[ y \geq 0 \][/tex]
