Question

If ƒ(x) is continuous and increasing function such that the domain of g(x) = $\lim_{x \rightarrow 2}f(x)$ be R and h (x) = $\lim_{x \rightarrow \pi/2}\frac{a^{\cot x} - a^{\cos x}}{\cot x - \cos x} =$ then the domain of φ(x) =$\log a$is –

• A. R
• B. {0, 1}
• C. R – {0, 1}
• D. R⁺ – {1}

Answer 1

Answer: (C)

R – {0, 1}

Answer 2

h(x) = $\frac { 1 } { 1 - x }$ , x ≠ 1 ⇒ h (h(x)) = $\frac { x - 1 } { x }$ , x ≠ 0, 1

∴ h(h(h(x))) = x, x ≠ 0, 1

Also, g(x) ≥ 0 ∀ x ∈ R ⇒ f(x) ≥ x ⇒ f(f(x)) ≥ f(x) ≥ x

[Q f(x) is increasing]

⇒ f(f(f(x))) ≥ f(x) ≥ x ⇒ f(f(f(x))) – x ≥ 0

∀ x ∈ R – {0, 1}

Q φ(x) is defined for all x ∈ R – {0, 1}.