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Linear Functions

SAT Math· difficulty 5/5

Let f(x)=2x−3f(x) = 2x - 3 and g(x)=ax+bg(x) = ax + b. If f(g(x))=g(f(x))f(g(x)) = g(f(x)) for all xx, which is necessarily true?

  • A

    a+b=−3a + b = -3

  • B

    b=−3b = -3

  • C

    2b−3a=b−32b - 3a = b - 3

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  • D

    a=1a = 1

Explanation

f(g(x))=2(ax+b)−3=2ax+2b−3f(g(x)) = 2(ax + b) - 3 = 2ax + 2b - 3. g(f(x))=a(2x−3)+b=2ax−3a+bg(f(x)) = a(2x - 3) + b = 2ax - 3a + b. Equal coefficients of xx match. Constants: 2b−3=−3a+b2b - 3 = -3a + b, i.e., 2b−3a=b−32b - 3a = b - 3.

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