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Linear Functions

SAT Math· difficulty 4/5

Let f(x)=mxf(x) = mx and g(x)=x+ng(x) = x + n. If f(g(x))=g(f(x))f(g(x)) = g(f(x)) for all xx, what must be true?

  • A

    m=1m = 1 or n=0n = 0

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  • B

    n=1n = 1 only

  • C

    m=0m = 0 only

  • D

    All m,nm, n

Explanation

f(g(x))=m(x+n)=mx+mnf(g(x)) = m(x + n) = mx + mn. g(f(x))=mx+ng(f(x)) = mx + n. Equal iff mn=nmn = n, i.e., n(m−1)=0n(m - 1) = 0.

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