Sure, let's find the range of the function [tex]\( y = \sqrt{x+7} + 5 \)[/tex] step-by-step.
1. Understand the function structure:
- We have a square root function [tex]\( \sqrt{x+7} \)[/tex].
- To which we add 5.
2. Square root function properties:
- The square root function [tex]\( \sqrt{x+7} \)[/tex] is only defined for [tex]\( x+7 \geq 0 \)[/tex], i.e., [tex]\( x \geq -7 \)[/tex].
- The smallest value [tex]\( \sqrt{x+7} \)[/tex] can take is 0, which happens when [tex]\( x = -7 \)[/tex].
- For [tex]\( x > -7 \)[/tex], [tex]\( \sqrt{x+7} \)[/tex] is positive.
3. Combining with the addition:
- The minimum value of [tex]\( \sqrt{x+7} \)[/tex] is 0.
- When [tex]\( \sqrt{x+7} = 0 \)[/tex], [tex]\( y = 0 + 5 = 5 \)[/tex].
4. Values greater than the minimum:
- For [tex]\( x > -7 \)[/tex], [tex]\( \sqrt{x+7} \)[/tex] will be greater than 0, so [tex]\( \sqrt{x+7} + 5 \)[/tex] will be greater than 5.
5. Conclusion about the range:
- The minimum value of [tex]\( y \)[/tex] is 5 and it gets larger the larger [tex]\( x \)[/tex] gets.
- Therefore, [tex]\( y \geq 5 \)[/tex].
So, the correct answer is:
[tex]\[ y \geq 5 \][/tex]
