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MCQEasyJEE 2024Composition & Inverse Functions

JEE Mathematics 2024 Question with Solution

If f(x)=4x+36x−4f(x) = \frac{4x + 3}{6x - 4}, with x≠23x \ne \frac{2}{3}, and (f∘f)(x)=g(x)(f \circ f)(x) = g(x), where g:R−{23}→R−{23}g: R - \left\{\frac{2}{3}\right\} \to R - \left\{\frac{2}{3}\right\}, then (g∘g∘g)(4)(g \circ g \circ g)(4) is equal to:

  • A

    1920\frac{19}{20}

  • B

    1920\frac{19}{20}

  • C

    −4-4

  • D

    44

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: f(x)=4x+36x−4f(x) = \frac{4x + 3}{6x - 4} and g(x)=(f∘f)(x)g(x) = (f \circ f)(x).

Find: (g∘g∘g)(4)(g \circ g \circ g)(4).

From the valid solution approach, composing ff with itself gives the identity transformation:

g(x)=f(f(x))=xg(x) = f(f(x)) = x

Now apply gg repeatedly at x=4x = 4:

g(4)=4g(4) = 4 g(g(4))=g(4)=4g(g(4)) = g(4) = 4 g(g(g(4)))=4g(g(g(4))) = 4

Therefore, the value of (g∘g∘g)(4)(g \circ g \circ g)(4) is 44. Hence, the correct option is D.

The other solution approach contains inconsistent working and uses a different function, so it is disregarded.

Common mistakes

  • Computing f(f(x))f(f(x)) using a different function such as f(x)=4x+56x−4f(x) = \frac{4x+5}{6x-4}. This is wrong because it does not match the given question. Always compose using the exact function stated in the question.

  • Treating g(x)g(x) as a new unrelated function instead of g(x)=(f∘f)(x)g(x) = (f \circ f)(x). This is wrong because the question explicitly defines gg through composition. First find f(f(x))f(f(x)), then iterate gg.

  • Stopping after finding g(4)g(4) and not evaluating g(g(g(4)))g(g(g(4))). This is wrong because three compositions are required. After identifying that g(x)=xg(x)=x, apply it three times carefully.

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