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Nonlinear Functions

SAT Math· difficulty 5/5

If g(x)=x+1g(x) = x + 1 and f(g(x))=x2+2x+5f(g(x)) = x^2 + 2x + 5, what is f(x)f(x)?

  • A

    x2+4x^2 + 4

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  • B

    x2+1x^2 + 1

  • C

    x2+5x^2 + 5

  • D

    x2+2x+4x^2 + 2x + 4

Explanation

Let u=g(x)=x+1u = g(x) = x + 1, so x=u−1x = u - 1. Then f(u)=(u−1)2+2(u−1)+5=u2−2u+1+2u−2+5=u2+4f(u) = (u - 1)^2 + 2(u - 1) + 5 = u^2 - 2u + 1 + 2u - 2 + 5 = u^2 + 4. So f(x)=x2+4f(x) = x^2 + 4.

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