If f(a) = a - 5 and g(b)= 5 − b, what is the value of the expression ।f(x)।− ।g(x)। + ।f[g(x)]।?

Question: If f(a) = a - 5 and g(b)= 5 − b, what is the value of the expression ।f(x)।− ।g(x)। + ।f[g(x)]।?

Solution:
f(x) = x - 5 ⇒ so |f(x)| = |x - 5|
g(x) = 5 - x ⇒ so |g(x)| = |5 - x| = |- (x - 5)| = |x - 5|

f(g(x)) = f(5 - x) = (5 - x) - 5 = - x ⇒ so |f(g(x))| = |- x|

Now plug into the expression:

|f(x)| - g(x)| + |f(g(x))| = |x - 5| - |x - 5| + |- x| = 0 + |- x| = |- x| = |x|

So the correct answer is গ) |- x|.
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Note: Why not ঙ) x ?

Actually, |- x| is always the same as |x|, not the same as the plain x unless x ≥ 0.
For a negative x (say, x = −2), we get
|−(−2)| = |2| = 2,
while x = −2.
cannot reduce to the bare x for all x.

So, Option ঙ) x would only match when x happens to be non-negative, but it fails for negative values.

That is why the correct answer is গ) ।- x।