Which of the following functions is/are identical to `|x-2|` ?

Options

A. (a) `f(x)=sqrt(x^(2)-4x+4) "`
B. (b) ` g(x)=|x|-|2|`
C. (c ) `h(x)=(|x-2|^(2))/(|x-2|) " `
D. (d) `"t(x)=|(x^(2)-x-2)/(x+1)|`

Answer

a

Explanation

`f(x)=sqrt(x^(2)-4x+4)=sqrt((x-2)^(2))=|x-2|`
`g(x)=|x|-|2|=|x|-2=={(-x-2",",x lt 2),(x-2",",x ge2):}`.
Thus `g(x)` is not same as `|x-2|`
`h(x)=(|h-2|^(2))/(|x-2|)=|x-2|,x ne 2.`
This is not same as `|x-2|` as `h(x)` has domain `R-{2}`
`t(x)=|(x^(2)-x-2)/(x+1)|=|((x-2)(x+1))/(x+1)|=|x-2|, x ne -1`.
Thus `t(x)` is not same as `|x-2|` as `t(x)` has domain `R-{-1}.`