Let #g(x)=sqrt(25-(x-2)^2)+3#
What is under the #sqrt# sign is #>=0#. this is the domain
So,
#25-(x-2)^2>=0#
#25-(x^2-4x+4)>=0#
#x^2-4x+4-25<=0#
#x^2-4x-21<=0#
Let's factorise
#(x-7)(x+3)<=0#
Let #f(x)=(x-7)(x+3)#
Let 's do a sign chart to solve this inequality
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-3##color(white)(aaaa)##7##color(white)(aaaa)##+oo#
#color(white)(aaaa)##x+3##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-#
#color(white)(aaaa)##x-7##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##-#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore,
#f(x)<=0# when #x in [-3,7]#, this is the domain
To calculate the range,
When #x=-3#, #=>#, #g(-3)=3#
When #x=7#, #=>#, #g(7)=3#
When #x=2#, #=>#, #g(2)=8#
Let #y=sqrt(25-(x-2)^2)+3#
The range is #y in [3,8]#
graph{(sqrt(25-(x-2)^2)+3) [-9.74, 12.76, -2.055, 9.195]}
