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Composite Linear Function Coefficients

XAT 2017 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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If f(x)=ax+bf(x) = ax + b, aa and bb are positive real numbers and if f(f(x))=9x+8f(f(x)) = 9x + 8, then the value of a+ba + b is:

  1. A.

    3

  2. B.

    4

  3. C.

    5

  4. D.

    6

  5. E.

    None of the above

Answer

C

Explanation

f(f(x))=a(ax+b)+b=a2x+ab+bf(f(x)) = a(ax + b) + b = a^2 x + ab + b. Given f(f(x))=9x+8f(f(x)) = 9x + 8, we equate coefficients: a2=9  ⟹  a=3a^2 = 9 \implies a = 3 (since aa is positive). ab+b=8  ⟹  3b+b=8  ⟹  4b=8  ⟹  b=2ab + b = 8 \implies 3b + b = 8 \implies 4b = 8 \implies b = 2.

Hence, a+b=3+2=5a + b = 3 + 2 = 5.

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