Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Functions question

2003 · Shift 0 · Q66
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Functions
  5. /2003 · Shift 0 · Q66

Functions question

2003 · Shift 0 · Q66

JEE MainMathematicsFunctionsMCQ+4 / −1
The function f(x)=log⁡(x+x2+1)f\left( x \right) = \log \left( {x + \sqrt {{x^2} + 1} } \right)f(x)=log(x+x2+1​), is
  1. A
    neither an even nor an odd function
  2. B
    an even function
  3. C
    an odd function
  4. D
    a periodic function
View written solutionFree

Correct answer: C

  1. We are given f(x)=log⁡(x+x2+1).f(x)=\log\left(x+\sqrt{x^2+1}\right).f(x)=log(x+x2+1​). We need to determine whether this function is even, odd, neither, or periodic.

  2. Recall:

  • A function is even if f(−x)=f(x).f(-x)=f(x).f(−x)=f(x).
  • A function is odd if f(−x)=−f(x).f(-x)=-f(x).f(−x)=−f(x).

  1. Compute f(−x)f(-x)f(−x): f(−x)=log⁡(−x+x2+1).f(-x)=\log\left(-x+\sqrt{x^2+1}\right).f(−x)=log(−x+x2+1​).

  2. Now simplify using the identity (x2+1+x)(x2+1−x)=x2+1−x2=1.\left(\sqrt{x^2+1}+x\right)\left(\sqrt{x^2+1}-x\right)=x^2+1-x^2=1.(x2+1​+x)(x2+1​−x)=x2+1−x2=1. So, x2+1−x=1x2+1+x.\sqrt{x^2+1}-x=\frac{1}{\sqrt{x^2+1}+x}.x2+1​−x=x2+1​+x1​. Hence, −x+x2+1=1x+x2+1.-x+\sqrt{x^2+1}=\frac{1}{x+\sqrt{x^2+1}}.−x+x2+1​=x+x2+1​1​. Therefore, f(−x)=log⁡(1x+x2+1).f(-x)=\log\left(\frac{1}{x+\sqrt{x^2+1}}\right).f(−x)=log(x+x2+1​1​).

  3. Use the logarithm property: log⁡(1a)=−log⁡(a).\log\left(\frac{1}{a}\right)=-\log(a).log(a1​)=−log(a). Thus, f(−x)=−log⁡(x+x2+1)=−f(x).f(-x)=-\log\left(x+\sqrt{x^2+1}\right)=-f(x).f(−x)=−log(x+x2+1​)=−f(x).

  4. Therefore, f(−x)=−f(x),f(-x)=-f(x),f(−x)=−f(x), so f(x)f(x)f(x) is an odd function.

  5. Checking options:

  • A: neither even nor odd ❌
  • B: even ❌
  • C: odd ✅
  • D: periodic ❌

So the correct option is C.

PreviousNext

More from Functions

  • A function f from the set of natural numbers to integers defined by f(n)={2n−1​,whennisodd−2n​,whenniseven​ is2003 · MCQ
  • If f:R→R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7, then r=1∑n​f(r) is2003 · MCQ
  • Domain of definition of the function f(x) = 4−x23​+log10​(x3−x), is2003 · MCQ
  • The domain of sin−1[log3​(3x​)] is2002 · MCQ
  • The period of sin2θ is2002 · MCQ
  • Which one is not periodic?2002 · MCQ
  • If the domain of the function f(x)=10+3x−x2​1​+x+∣x∣​1​ is (a,b), then (1+a)2+b2 is equal to :2025 · MCQ
  •  If the domain of the function f(x)=loge​(5+4x2x−3​)+sin−1(2−x4+3x​) is [α,β), then α2+4β is equal to 2025 · MCQ