The domain of `f(x)=cos^(-1)((2-|x|)/4)+[ log(3-x)]^-1` is (a)`[-2,6]` (b) `[-6,2)uu(2,3)` (c) `[-6,2]` (d) `[-2,2]uu(2,3)`

Options

A. `[-2,6]`
B. `[-6,2) cup (2,3)`
C. `[-6,2]`
D. `[-2,2] cup (2,3)`

Answer

B

Explanation

` "cos"^(-1)((2-|x|)/(4))` exists if
`-1 le (2-|x|)/(4) le 1`
or `-6 le -|x| le 2`
or `-2 le |x| le 6`
or `|x| le 6`
or `-6 le x le 6`
The function `[log(3-x)]^(-1)=(1)/(log(3-x))` is defined if `3-x gt 0 " and " x ne 2," i.e., if " x ne 2 " and " x lt 3.`
Thus, the domain of the given function is
`{x|-6 le x le 6} cap {x| x ne 2, x lt 3}=[-6,2) cup (2,3)`