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Functions: Domain Definition Denotes Greatest Integer Function

JEE Maths question with a full step-by-step solution.

Question

Domain of the definition

f(x)=\dfrac{\log_{4}(5-[x-1]-[x]^{2})}{x^{2}+x-2}

(where

[\cdot]

denotes greatest integer function) is:

(-3,-2)\cup(1,2)

(-4,-3)\cup(2,3)\cup(3,4)

\left(-\dfrac{4}{3},-\dfrac{3}{4}\right)\cup\left(\dfrac{2}{3},\dfrac{2}{3}\right)

(-2,1)\cup(1,2)

correct

Solution

Step 1: For

f(x)

to be defined, two conditions must hold simultaneously. Condition 1: Denominator

\neq 0

x^{2}+x-2\neq 0\;\Rightarrow\;(x+2)(x-1)\neq 0\;\Rightarrow\;x\neq -2,\;x\neq 1

Condition 2: Argument of

\log_{4}

must be

>0

5-[x-1]-[x]^{2}>0

Step 2: Note that

[x-1]=[x]-1

for all

x

. Substituting:

5-([x]-1)-[x]^{2}>0\;\Rightarrow\;6-[x]-[x]^{2}>0\;\Rightarrow\;[x]^{2}+[x]-6<0

Step 3: Solve

[x]^{2}+[x]-6<0

([x]+3)([x]-2)<0\;\Rightarrow\;-3<[x]<2

[x]\in\{-2,-1,0,1\}

, giving

x\in[-2,2)

. Step 4: Combine with Condition 1 (

x\neq -2,\;x\neq 1

x\in(-2,2)\setminus\{1\}=(-2,1)\cup(1,2)

Answer: (4)

(-2,1)\cup(1,2)

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