The domain of the function fx-x2-x-2 is - represents the greatest integer function

F(x)=√(x^2 [x]^2) f is defined if nbsp; x^2 [x]^2≥0 ⋯(1) ⇒ (x [x])(x+[x])≥0 ⇒ x+[x]≥0 (∵ x [x]≥ 0 ∀ x∈ℝ) ⇒ x≥0 Also, if x ∈ℤ,x=[x] So, eqn(1) is sati ...

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The domain of...

Question

The domain of the function $f (x) = \sqrt{x^{2} - [x]^{2}}$ is
$($ $[.]$ represents the greatest integer function $)$

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Solution

$f (x) = \sqrt{x^{2} - [x]^{2}}$
$f$ is defined if
$x^{2} - [x]^{2} \geq 0 \dots (1)$
$\Rightarrow (x - [x]) (x + [x]) \geq 0$
$\Rightarrow x + [x] \geq 0 (∵ x - [x] \geq 0 \forall x \in R)$
$\Rightarrow x \geq 0$

Also, if $x \in Z, x = [x]$
So, eqn $(1)$ is satisfied for $x \in Z$ also.

Hence, the domain of $f$ is ${x : x \geq 0} \cup Z$

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The domain of the function f(x)=√x2−[x]2 is ( [.] represents the greatest integer function )

f(x)=√x2−[x]2 f is defined if x2−[x]2≥0 ⋯(1) ⇒(x−[x])(x+[x])≥0 ⇒x+[x]≥0 (∵x−[x]≥0 ∀ x∈R) ⇒x≥0 Also, if x∈Z,x=[x] So, eqn(1) is satisfied for x∈Z also. Hence, the domain of f is {x:x≥0}∪Z

The domain of the function $f (x) = \sqrt{x^{2} - [x]^{2}}$ is
$($ $[.]$ represents the greatest integer function $)$