To determine the range of the function [tex]\( y = \sqrt{x + 7} + 5 \)[/tex], let's examine the behavior of the function step by step.
1. Understand the Function:
The given function is [tex]\( y = \sqrt{x + 7} + 5 \)[/tex]. Here, the term [tex]\(\sqrt{x + 7}\)[/tex] is a square root function, and it is crucial to understand its properties.
2. Domain of the Square Root Function:
The square root function [tex]\(\sqrt{x + 7}\)[/tex] is defined for values where the expression inside the square root is non-negative. Therefore:
[tex]\[ x + 7 \geq 0 \][/tex]
[tex]\[ x \geq -7 \][/tex]
This means the function [tex]\( \sqrt{x + 7} \)[/tex] is defined for [tex]\( x \geq -7 \)[/tex].
3. Behavior of the Square Root Function:
The square root function, [tex]\( \sqrt{x + 7} \)[/tex], produces non-negative values:
[tex]\[ \sqrt{x + 7} \geq 0 \][/tex]
This is because the square root of any non-negative number is non-negative.
4. Adding a Constant:
The given function adds 5 to the square root function:
[tex]\[ y = \sqrt{x + 7} + 5 \][/tex]
Since [tex]\(\sqrt{x + 7}\)[/tex] is always non-negative, adding 5 ensures that [tex]\( y \)[/tex] is always greater than or equal to 5:
[tex]\[ \sqrt{x + 7} \geq 0 \][/tex]
Therefore:
[tex]\[ \sqrt{x + 7} + 5 \geq 0 + 5 \][/tex]
[tex]\[ y \geq 5 \][/tex]
5. Range of the Function:
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x + 7} \)[/tex] also increases, which means [tex]\( y \)[/tex] increases without any upper limit. This implies that:
[tex]\[ y \) can take any value greater than or equal to 5. \][/tex]
6. Conclusion:
Hence, the range of [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is:
[tex]\[ y \geq 5 \][/tex]
Based on the given choices, the correct answer is:
[tex]\[ y \geq 5 \][/tex]
So, the range of [tex]\( y = \sqrt{x + 7} + 5 \)[/tex] is [tex]\( y \geq 5 \)[/tex].
