Let \( f(x)=|x-1| \), Then Which Statement is Correct?
Question:
Let \( f(x)=|x-1| \). Then,
(a) \( f(x^2)=[f(x)]^2 \)
(b) \( f(x+y)=f(x)f(y) \)
(c) \( f(|x|)=|f(x)| \)
(d) none of these
Solution:
Check option (c):
\[ f(|x|)=||x|-1| \]
and
\[ |f(x)|=||x-1|| \]
These are not equal in general.
Check option (a):
\[ f(x^2)=|x^2-1| \]
\[ [f(x)]^2=(|x-1|)^2=(x-1)^2 \]
Not equal.
Check option (b):
\[ f(x+y)=|x+y-1| \]
\[ f(x)f(y)=|x-1||y-1| \]
Not equal.
Hence,
\[ \boxed{\text{Correct Answer: (d) none of these}} \]