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)xcolor(white)(aaaa)-oocolor(white)(aaaa)-3color(white)(aaaa)7color(white)(aaaa)+oo
color(white)(aaaa)x+3color(white)(aaaaa)-color(white)(aaaa)+color(white)(aaaa)-
color(white)(aaaa)x-7color(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]}
