Question

For x ∈ (0, 3/2), let f(x) = √x, g(x) = tan x and h(x) = (1-x²)/(1+x²). If Φ(x) = ((hof)og)(x), then Φ(π/3) is equal to :
(A) tan(11π/12)
(B) tan (π/12)
(C) tan (5π/12)
(D) tan (7π/12)

Answer 1

Answer:

option A...................

Answer 2

value of Φ(π/3) = tan(11π/12)

•Given,

f(x) = √x

g(x) = tan x

h(x) = (1-x²)/(1+x²)

•(hof) = h[f(x)]

= h(√x)

= [1-(√x)²]/[1+(√x)²]

= (1-x)/(1+x)

•((hof)og) = (hof)[g(x)]

=(hof)[tanx]

= (1-tanx)(1+tanx)

•Φ(x) = ((hof)og)(x)

Φ(x) = (1-tanx)(1+tanx)

•Φ(π/3) =[ 1-tan(π/3)]/(1+tan(π/3)]

Φ(π/3) =[ 1-√3]/[1+√3]

Φ(π/3) = [1-√3]²/[1+√3][1-√3]

Φ(π/3) = (1+3-2√3)/(1-3)

Φ(π/3) = (4-2√3)/(-2)

Φ(π/3) = √3-2

= tan(11π/12)

•hence option A is correct