Function

a)f(x) = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ)y = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ)y = eˣ (eˣ - e⁻ˣ) / (eˣ (eˣ + e⁻ˣ))y = (e²ˣ - 1) / (e²ˣ + 1)(y - 1) e²ˣ + y + 1 = 0e²ˣ = (1 + y) / (1 - y) > 0 , i.e. (1 + y) (1 - y) > 0 , ==> - 1 < y < 12x = In ( (1 + y) / (1 - y) )x = (1/2) In ( (1 + y) / (1 - y) )So f⁻¹ (x) = (1/2) In ( (1 + x) / (1 - x) ) , (- 1 < x < 1)
b)g ( f(x) ) = In f(x)= In ( (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ) )= In ( (e²ˣ - 1) / (e²ˣ + 1) )So (e²ˣ - 1) / (e²ˣ + 1) > 0 (e²ˣ - 1) (e²ˣ + 1) > 0e ⁴ˣ > 14x > In 1 = 0x > 0Domain of g ( f(x) ) is all positive real numbers.