Question

The domain of f(x) = $\sqrt{\log_{\frac{1}{4}}\left( \frac{5x - x^{2}}{4} \right)} ⥄ ⥄$+ ¹⁰C_x is

• A. (0, 1]U [4, 5)
• B. (0, 5)
• C. {1, 4}
• D. None of these

Answer 1

Answer: (C)

{1, 4}

Answer 2

Let f₁ = $\sqrt{\log_{\frac{1}{4}}\left( \frac{5x - x^{2}}{4} \right)}$ and f₂ = ¹⁰C_x.

Clearly f₁ is defined for $\log_{\frac{1}{4}}\left( \frac{5x - x^{2}}{4} \right)$ ≥ 0

⇒ 0 < $\frac{5x - x^{2}}{4}$ ≤ 1

⇒ $\frac{5x - x^{2}}{4}$ > 0 and $\frac{5x - x^{2}}{4}$ ≤ 1

⇒ x (x – 5) < 0 and x² – 5x + 4 ≥ 0

⇒ x$\in$ (0, 5) and x$\in$ (-$\infty$, 1]U [4, $\infty$)

⇒ f₁ is defined for x$\in$ (0, 1] U [4, 5) and f₂ is defined for x $\in${0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ⇒ f(x) is defined for x$\in$D_f1 $\cap$ D_f2 = {1, 4}