Let `a=sin^(-)(sin3)+sin^(-1)(sin4)+sin^(-1)(sin5),f(x)=e^(x^(2)+|x|),` domain of `f(x)` be `[a,oo)` & range of `f(x)` be `[b,oo)` and `g(x)=(4cos^(4)x-2cos2x-1/4"cos"4x-x^(7))^(1//7)`, domain & range of `g(x)` is set of real numbers. Which of the following are correct

Options

A. `a=-2`
B. `a+b=-1`
C. `f(g(g(b))=e^(2)`
D. Both `f(x), g(x)` are non invertible fns

Answer

A, B, C

Explanation

`a=(pi-3)=(pi-4)+(5-2pi)=-2`
`f(-2)=f(2)impliesf(x)` is many one `implies` non invertible
Let `t=x^(2)+|x|,t epsilon [0,oo)`
`f(x) epsilon [1,oo)`
`impliesb=1` & `a+b=-1`
`g(x)=[(1+cos2x)^(2)-2cosx-1/2(2cos^(2)2x-1)-x^(7)]^(1//7)`
`g(x)=(3/2-x^(7)]^(1/7)`
`f(g(g(b)))=f(b)=e^(2)`