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Given: f(x) = √(tan²x - 3)
x ∈ [-4π , -3π]
To Find : Values of x which f(x) is defined
Solution:
f(x) = √(tan²x - 3)
f(x) is defined for all the domains of tan(x)
if tan²x - 3 ≥ 0
Tan x is not defined for multiples of π/2
Hence -7π/2 is not domain of tan(x) for x ∈ [-4π , -3π]
Now checking tan²x - 3 ≥ 0
=> tanx ≥ √3 or tanx ≤ -√3
in principal values
x ∈ [ π/3 , π/2) and x ∈ (-π/2 , -π/3]
tan x = tan(nπ+x)
x ∈ [ π/3 , π/2) can be x ∈ [ π/3 -4π , π/2 -4π)
=> x ∈ [ -11π/3 , -7π/2)
x ∈ (-π/2 , -π/3] can be x ∈ (-π/2 -3π , -π/3 - 3π]
=> x ∈ ( -7π/2 , -10π/3]
x ∈ [ -11π/3 , -7π/2) and x ∈ ( -7π/2 , -10π/3]
=> x ∈ [ -11π/3 , -10π/3) - { -7π/2}
which f(x) is defined for x ∈ [ -11π/3 , -10π/3) - { -7π/2}
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