FunctionHard
Question
Which of following pairs of functions are identical.
Options
A.f(x) = eln sec-1x and g(x) = sec-1 x
B.f(x) = tan (tan-1 x) and g(x) = cot (cot-1 x)
C.f(x) = sgn (x) and g(x) = sgn (sgn (x))
D. f(x) = cot2 x. cos2 x and g(x) = cot2 x - cos2 x
Solution
(A) f(x) = eln(sec-1x) = sec-1x, x ∈ (-∞, -1] ∪ (1, ∞)
g(x) = sec-1x, x ∈(-∞, -1] ∪ [1, ∞)
non-identical functions
(B) f(x) = tan (tan-1 x) = x, x ∈ R
g(x) = cot (cot-1 x) = x, x ∈ R
identical functions
(C) f(x) = sgn (x) =
g(x) = sgn(sgn x) =
Identical functions
(D) f(x) = cot2 x . cos2 x, x ∈ R - {n π}, n ∈ I
g(x) = cot2 x - cos2 x
= cot2 x (1 - sin2 x)
= cot2 x. cos2 x x∈R - {n π}, n ∈ I
Identical functions
g(x) = sec-1x, x ∈(-∞, -1] ∪ [1, ∞)
non-identical functions
(B) f(x) = tan (tan-1 x) = x, x ∈ R
g(x) = cot (cot-1 x) = x, x ∈ R
identical functions
(C) f(x) = sgn (x) =
g(x) = sgn(sgn x) =
Identical functions
(D) f(x) = cot2 x . cos2 x, x ∈ R - {n π}, n ∈ I
g(x) = cot2 x - cos2 x
= cot2 x (1 - sin2 x)
= cot2 x. cos2 x x∈R - {n π}, n ∈ I
Identical functions
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