Find the domain and range of h(x) = g(f(x), where
`f(x) = {([x], -2lexle-1),(|x| + 1, -1 lt x le 2) :}` and `g(x) = {([x], -pi le x le 0), (sin x, 0 lt x le pi):}`

Options

A. `[sin 3, sin 1]`
B. `[sin3,1] cup {-2,-1,0}`
C. `[sin3,1] cup {-2, -1}`
D. `[sin1,1]`

Answer

C

Explanation

`g(f(x))={([f(x)]",",-pi le f(x) lt 0),(sin f(x)",",0 le f(x) le pi):}`
`={([[x]]",",-pi le [x] lt 0",",-2 le x le -1),([|x|+1]",",-pi le |x| +1 lt 0",",-1 lt x le 2),(sin[x]",",0 le [x] le pi",",-2 le x le -1),(sin(|x|+1)",",0le |x|+1 le pi",",-1 lt x le 2):}`
`={([x]",",-2 le x le -1),(sin(|x|+1)",", -1 lt x le 2):}`
Hence, the domain is [-2, 2].
Also, for `-2 le x le -1, [x]= -2, -1,`
and for `-1 lt x le 2, |x| +1 in [1,3]`
` :. sin (|x|+1) in [sin 3,1]`
Hence, the range is `{-2, -1} cup [sin 3,1].`
Also, for `y in [sin 3,1], [y]=0,1.`
Hence, the number of integral points in the range is 4.