FunctionsmediumFree

Functions: Domain Function

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

Question

The domain of the function

y=\sqrt\dfrac{2x-1}{2x^{3}+3x^{2}+x}+\sqrt{\sin^{-1}(\log_{2}x)}

is:

\left(\dfrac{1}{2},\infty\right)

\left(\dfrac{1}{2},2\right]

[1,2]

correct

(1,\infty)

Solution

Solution: Step 1: Condition (i):

x>0

(for

\log_{2}x

to be defined). Condition (ii):

\sin^{-1}(\log_{2}x)

requires

-1\le\log_{2}x\le 1\;\Rightarrow\;2^{-1}\le x\le 2^{1}

, so

x\in\left[\dfrac{1}{2},2\right]\quad\ldots(2)

Condition (iii):

\sin^{-1}(\log_{2}x)\ge 0

(needed for outer square root):

\log_{2}x\ge\sin 0=0\;\Rightarrow\;x\ge 1\quad\ldots(3)

Condition (iv):

\dfrac{2x-1}{2x^{3}+3x^{2}+x}\ge 0

. Factor denominator:

x(2x+1)(x+1)

. For

x>0

, denominator

>0

. So need numerator

\ge 0

2x-1\ge 0\;\Rightarrow\;x\ge\dfrac{1}{2}\quad\ldots(4)

Also denominator

\neq 0

x\neq 0,-\dfrac{1}{2},-1

(all satisfied for

x\ge\dfrac{1}{2}>0

). Step 2: Combining (1), (2), (3), (4):

x\in[1,2]

. Answer: (C)

[1,2]

Still stuck on this question?Ask your doubt on WhatsApp

Solve more, learn faster

Sign up free to solve more JEE Mathsquestions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.