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Functions question

2020 · 7 Jan · Shift 1 · Q37
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Functions question

2020 · 7 Jan · Shift 1 · Q37

JEE MainMathematicsFunctionsMCQ+4 / −1
If g(x) = x2 + x - 1 and (goƒ) (x) = 4x2 - 10x + 5, then ƒ(54)\left( {{5 \over 4}} \right)(45​) is equal to:
  1. A
    12{1 \over 2}21​
  2. B
    32{3 \over 2}23​
  3. C
    -12{1 \over 2}21​
  4. D
    -32{3 \over 2}23​
View written solutionFree

Correct answer: C

  1. We are given g(x)=x2+x−1g(x)=x^2+x-1g(x)=x2+x−1 and (g∘f)(x)=g(f(x))=4x2−10x+5.(g\circ f)(x)=g(f(x))=4x^2-10x+5.(g∘f)(x)=g(f(x))=4x2−10x+5.

  2. Substitute f(x)f(x)f(x) into ggg: g(f(x))=(f(x))2+f(x)−1.g(f(x))=(f(x))^2+f(x)-1.g(f(x))=(f(x))2+f(x)−1. Hence, (f(x))2+f(x)−1=4x2−10x+5.(f(x))^2+f(x)-1=4x^2-10x+5.(f(x))2+f(x)−1=4x2−10x+5.

  3. Rearrange: (f(x))2+f(x)=4x2−10x+6.(f(x))^2+f(x)=4x^2-10x+6.(f(x))2+f(x)=4x2−10x+6. Factor the right-hand side: 4x2−10x+6=(2x−2)(2x−3).4x^2-10x+6=(2x-2)(2x-3).4x2−10x+6=(2x−2)(2x−3). Also, 4x2−10x+6=(2x−3)2+(2x−3).4x^2-10x+6=(2x-3)^2+(2x-3).4x2−10x+6=(2x−3)2+(2x−3). because (2x−3)2+(2x−3)=4x2−12x+9+2x−3=4x2−10x+6.(2x-3)^2+(2x-3)=4x^2-12x+9+2x-3=4x^2-10x+6.(2x−3)2+(2x−3)=4x2−12x+9+2x−3=4x2−10x+6.

  4. So we have (f(x))2+f(x)=(2x−3)2+(2x−3).(f(x))^2+f(x)=(2x-3)^2+(2x-3).(f(x))2+f(x)=(2x−3)2+(2x−3). This suggests f(x)=2x−3f(x)=2x-3f(x)=2x−3 or f(x)=−1−(2x−3)=2−2x,f(x)=-1-(2x-3)=2-2x,f(x)=−1−(2x−3)=2−2x, since if y2+y=t2+ty^2+y=t^2+ty2+y=t2+t, then either y=ty=ty=t or y=−1−ty=-1-ty=−1−t.

  5. We need f(54)f\left(\frac54\right)f(45​).

    • If f(x)=2x−3f(x)=2x-3f(x)=2x−3, then f(54)=2⋅54−3=52−3=−12.f\left(\frac54\right)=2\cdot\frac54-3=\frac52-3=-\frac12.f(45​)=2⋅45​−3=25​−3=−21​.

    • If f(x)=2−2xf(x)=2-2xf(x)=2−2x, then f(54)=2−2⋅54=2−52=−12.f\left(\frac54\right)=2-2\cdot\frac54=2-\frac52=-\frac12.f(45​)=2−2⋅45​=2−25​=−21​.

    In both cases, the value is the same.

  6. Therefore, f(54)=−12.f\left(\frac54\right)=-\frac12.f(45​)=−21​.

  7. Checking options:

    • A: 12\frac1221​
    • B: 32\frac3223​
    • C: −12-\frac12−21​ ✅
    • D: −32-\frac32−23​

So the correct option is C.

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