If `f: RvecS ,` defined by `f(x)=sinx-sqrt(3)cosx+1,is onto ,` then find the set `Sdot`

Options

A. `[-1//3]`
B. `[-1, 1 ]`
C. `[0, 1 ]`
D. `[0, 3]`

Answer

A

Explanation

If `f: R to S ` in onto, then range of f is equal to its co-domain i.e., S =Range (f).
Now,
`f(x) = sin x - sqrt(3) cos x+1`
`rArr f(x) = 2((1)/(2) sin x -(sqrt(3))/(2) cosx)+1`
`rArr f(x) = 2 sin (x-(pi)/(3)) +1`
We know that
`-1 le sin (x-(pi)/(3)) le 1"forall x" in R`
`rArr - 2 le 2 sin (x-(pi)/(3)) le "2 for all x" in R`
`rArr - 1le 2 sin (x-(pi)/(3))+ 1 le "3 for all x in R`
`rArr -1 le f(x) le 3 "for all x" in R`
Range (f) = [-1,3]
Hence, `S = [-1,3]`