What is the domain and range of #y = 5 - (sqrt(9-x^2))#?

1 Answer
Feb 26, 2018

Answer:

Donain: #[-3,+3]# Range: #[2, 5]#

Explanation:

#f(x) =5-(sqrt(9-x^2))#

#f(x)# is defined for #9-x^2>=0 -> x^2 <=9#

#:. f(x) # is defned for #absx <=3#

Hence the domain of #f(x)# is #[-3,+3]#

Consider, #0<= sqrt(9-x^2) <= 3# for #x in [-3,+3]#

#:.f_max = f(abs3) = 5-0 = 5#

and, #f_min = f(0) = 5 -3 =2#

Hence, the range of #f(x)# is #[2,5]#

We can see these results from the graph of #f(x)# below.

graph{5-(sqrt(9-x^2)) [-8.006, 7.804, -0.87, 7.03]}