Question

If $f\left(x\right)=|x-1|,$ then which of the following are true?

• A. $f\left(x^2\right)=\left(f\right(x){)}^2$
• B. $f(x+y)=f\left(x\right)+f\left(y\right)$
• C. $f\left(|x|\right)=|f\left(x\right)|$
• D. None of the above

Answer 1

The correct option is D None of the above

Given that $f\left(x\right)=|x-1|.$
$\Rightarrow f\left(x^2\right)=|x^2-1|=|x-1||x+1|$
But, $\left(f\right(x){)}^2=|x-1{|}^2=|x-1||x-1|\ne f(x^2).$

Now, $f(x+y)=|x+y-1|.$
Whereas, $f\left(x\right)+f\left(y\right)=|x-1|+|y-1|\ne f(x+y).$

Also, $f\left(|x|\right)=||x|-1|.$
But, $|f\left(x\right)|=|x-1|\ne f\left(|x|\right).$