we proceed to resolve each case to determine the solution
we know that
The radicand is a number or quantity from which a root is to be extracted
In this problem the radicand must be greater than or equal to zero
case A [tex]y=\sqrt{x+11} +5[/tex]
the radicand is equal to
[tex](x+11)[/tex]
so
[tex](x+11) \geq 0\\x \geq -11[/tex]
the domain is the interval--------> [-11.∞)
therefore
the function [tex]y=\sqrt{x+11} +5[/tex] is the solution of the problem
case B [tex]y=\sqrt{x-11} +5[/tex]
the radicand is equal to
[tex](x-11)[/tex]
so
[tex](x-11) \geq 0\\x \geq 11[/tex]
the domain is the interval--------> [11.∞)
therefore
the function [tex]y=\sqrt{x-11} +5[/tex] is not the solution of the problem
case C [tex]y=\sqrt{x+5}-11[/tex]
the radicand is equal to
[tex](x+5)[/tex]
so
[tex](x+5) \geq 0\\x \geq -5[/tex]
the domain is the interval--------> [-5.∞)
therefore
the function [tex]y=\sqrt{x+5} -11[/tex] is not the solution of the problem
case D [tex]y=\sqrt{x+5}+11[/tex]
the radicand is equal to
[tex](x+5)[/tex]
so
[tex](x+5) \geq 0\\x \geq -5[/tex]
the domain is the interval--------> [-5.∞)
therefore
the function [tex]y=\sqrt{x+5}+11[/tex] is not the solution of the problem
the answer is the function [tex]y=\sqrt{x+11} +5[/tex]
