FunctionHard
Question
Which one of the following pair of functions are identical ?
Options
A.e(lnx)/2 and √x
B.tan-1(tan x) and cot-1 (cot x)
C.cos2x + sin4x and sin2x + cos4x
D.
and sgn (x), where sgn(x) stands for signum function.
and sgn (x), where sgn(x) stands for signum function.Solution
(A) f(x) = e1/2 ln x = √x, D : x > 0
g(x) = √x, D : x ≥ 0
(B) tan-1 (tan x) = x D : x ≠ ± (2n + 1)
cot-1 (cot x) = x D : x ≠ ±nπ
(C) f(x) = cos2x + sin4x
= cos2x + (1 - cos2x)2
= 1 - cos2x + cos4x
= sin2x + cos4x
g(x) = sin2x + cos4x
(D) f(x) =
, D : x ≠ 0
g(x) = sgn (x), D : x ∈ R
g(x) = √x, D : x ≥ 0
(B) tan-1 (tan x) = x D : x ≠ ± (2n + 1)
cot-1 (cot x) = x D : x ≠ ±nπ
(C) f(x) = cos2x + sin4x
= cos2x + (1 - cos2x)2
= 1 - cos2x + cos4x
= sin2x + cos4x
g(x) = sin2x + cos4x
(D) f(x) =
, D : x ≠ 0g(x) = sgn (x), D : x ∈ R
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