Which one of the following is correct in respect of the function `f:RRtoRR^(+)` defined as `f(x)=|x+1|`?

Options

A. `f(x)^(2)=[f(x)]^(2)`
B. `f(|x|)=|f(x)|`
C. `f(x+y)=f(x)+f(y)`
D. None of the above

Answer

D

Explanation

Given, `f(x)=|x+1|`
Let us check all options.
(a) `f(x)^(2)=|x^(2)+1|`
`(f(x))^(2)=(x+1)^(2)`.
So, `f(x)^(2)ne(f(x))^(2)`
(b) `f(|x|)=||x|+1|`
`|f(x)|=||x+1||=|x+1|`
`f(x)ne|f(x)|`
(c) `f(x)+f(y)=|x+1|+|y+1|`
`f(x)+f(y)=|x+1|+|y+1|`
`f(x+y)nef(x+f(y))`