The range of the function in the first photo is
x-3>=0
sqrt(x-3)>=sqrt(0)
sqrt(x-3)>=0
Multiplying the equation by -2:
-2[ sqrt(x-3)>=0]
-2 sqrt(x-3)<=0
Adding 8 both sides of the equation:
-2 sqrt(x-3)+8<=0+8
-2 sqrt(x-3)+8<=8
f(x)<=8
Then Range f(x)= (-Infinite, 8]
The function has the same range is the third one:
g(x)=-sqrt(x+3)+8
The range of this function is:
x+3>=0
sqrt(x+3)>=sqrt(0)
sqrt(x+3)>=0
Multiplying the equation by -1:
-1[ sqrt(x+3)>=0]
- sqrt(x+3)<=0
Adding 8 both sides of the equation:
- sqrt(x+3)+8<=0+8
- sqrt(x+3)+8<=8
g(x)<=8
Then Range g(x)= (-Infinite, 8]
Answer: Third option: g(x)=-sqrt(x+3)+8