Question

The domain of definition of  \[f\left( x \right) = \sqrt{x - 3 - 2\sqrt{x - 4}} - \sqrt{x - 3 + 2\sqrt{x - 4}}\] is 

 

पर्याय

• (a) [4, ∞)

• (b) (−∞, 4]

• (c) (4, ∞)

• (d) (−∞, 4)

   

Answer 1

(a) [4, ∞)  \[f\left( x \right) = \sqrt{x - 3 - 2\sqrt{x - 4}} - \sqrt{x - 3 + 2\sqrt{x - 4}}\]

\[\text{ For f(x) to be defined } , x - 4 \geq 0\]

\[ \Rightarrow x - 4 \geq 0\]

\[ \Rightarrow x \geq 4 . . . . (1)\]

\[\text{ Also} , x - 3 - 2\sqrt{x - 4} \geq 0\]

\[ \Rightarrow x - 3 - 2\sqrt{x - 4} \geq 0\]

\[ \Rightarrow x - 3 \geq 2\sqrt{x - 4}\]

\[ \Rightarrow (x - 3 )^2 \geq \left( 2\sqrt{x - 4} \right)^2 \]

\[ \Rightarrow x^2 + 9 - 6x \geq 4\left( x - 4 \right)\]

\[ \Rightarrow x^2 - 10x + 25 \geq 0\]

\[ \Rightarrow (x - 5) {}^2 \geq 0, \text{ which is always true .}  \]

\[\text{ Similarly,}  x - 3 + 2\sqrt{x - 4} \geq 0 \text{ is always true } . \]

\[\text{ Thus, dom } (f(x)) = [4, \infty )\]