Find the domain and range of the real function `f(x) = sqrt(4-x^(2))`

Options

A. `(0,oo)`
B. `(-oo,oo)`
C. `(-oo, log_(e) 2]`
D. `(log_(e)2,oo)`

Answer

C

Explanation

Clearly , `f(x)=log_(e)sqrt(4-x^(2))` is defined for all ` x in (-2,2)`
Let y =f(x). Then,
`y=logesqrt(4-x^(2)) implies e^(2y)=4-x^(2)implies x=sqrt(4-e^(2y))`
Clearly , x is real , if
`4-e^(2y) ge 0 implies 0 lt e^(y) le 2 " "[ :. e^(y) gt 0]`
`implies -oo lt 6 le log_(e)2implies y in (-oo, log_(e)2]`