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Functions question

2019 · 9 Apr · Shift 2 · Q32
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Functions question

2019 · 9 Apr · Shift 2 · Q32

JEE MainMathematicsFunctionsMCQ+4 / −1
The domain of the definition of the function f(x)=14−x2+log⁡10(x3−x)f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)f(x)=4−x21​+log10​(x3−x) is
  1. A
    (-1, 0) ∪\cup∪ (1, 2) ∪\cup∪ (2, ∞\infty∞)
  2. B
    (-2, -1) ∪\cup∪ (-1,0) ∪\cup∪ (2, ∞\infty∞)
  3. C
    (1, 2) ∪\cup∪ (2, ∞\infty∞)
  4. D
    (-1, 0) ∪\cup∪ (1,2) ∪\cup∪ (3, ∞\infty∞)
View written solutionFree

Correct answer: A

  1. We need the function f(x)=14−x2+log⁡10(x3−x)f(x)=\frac{1}{4-x^2}+\log_{10}(x^3-x)f(x)=4−x21​+log10​(x3−x) to be defined.

So both parts must be defined simultaneously.

  1. Condition for the rational term: 14−x2\frac{1}{4-x^2}4−x21​ is defined when the denominator is nonzero.

So, 4−x2≠04-x^2\neq 04−x2=0 x2≠4x^2\neq 4x2=4 x≠±2x\neq \pm 2x=±2

  1. Condition for the logarithmic term: log⁡10(x3−x)\log_{10}(x^3-x)log10​(x3−x) is defined when its argument is positive: x3−x>0x^3-x>0x3−x>0

Factorize: x3−x=x(x2−1)=x(x−1)(x+1)x^3-x=x(x^2-1)=x(x-1)(x+1)x3−x=x(x2−1)=x(x−1)(x+1)

So we solve x(x−1)(x+1)>0x(x-1)(x+1)>0x(x−1)(x+1)>0

Critical points are x=−1,0,1x=-1,0,1x=−1,0,1.

Now check intervals:

  • For x<−1x<-1x<−1, take x=−2x=-2x=−2: (−)(−)(−)=−<0(-)(-)(-)=-<0(−)(−)(−)=−<0
  • For −1<x<0-1<x<0−1<x<0, take x=−12x=-\frac12x=−21​: (−)(−)(+)=+>0(-)(-)(+)=+>0(−)(−)(+)=+>0
  • For 0<x<10<x<10<x<1, take x=12x=\frac12x=21​: (+)(−)(+)=−<0(+)(-)(+)=-<0(+)(−)(+)=−<0
  • For x>1x>1x>1, take x=2x=2x=2: (+)(+)(+)=+>0(+)(+)(+)=+>0(+)(+)(+)=+>0

Hence, x3−x>0  ⟺  x∈(−1,0)∪(1,∞)x^3-x>0 \iff x\in (-1,0)\cup(1,\infty)x3−x>0⟺x∈(−1,0)∪(1,∞)

  1. Combine both conditions:

From log condition: x∈(−1,0)∪(1,∞)x\in (-1,0)\cup(1,\infty)x∈(−1,0)∪(1,∞)

Also exclude x=2x=2x=2 (and x=−2x=-2x=−2, though −2-2−2 is not in the above set anyway).

Therefore domain is (−1,0)∪(1,2)∪(2,∞)(-1,0)\cup(1,2)\cup(2,\infty)(−1,0)∪(1,2)∪(2,∞)

  1. Compare with options:

This matches Option A.

  1. Comparison with stored correct answer: Stored correct answer is A, which agrees with our result.
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