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Nonlinear Functions

SAT Math· difficulty 5/5

If f(x)=xx−1f(x) = \frac{x}{x - 1}, what is f−1(x)f^{-1}(x)?

  • A

    1−xx\frac{1 - x}{x}

  • B

    xx−1\frac{x}{x - 1}

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  • C

    x−1x\frac{x - 1}{x}

  • D

    x+1x\frac{x + 1}{x}

Explanation

y=xx−1⇒y(x−1)=x⇒yx−y=x⇒x(y−1)=y⇒x=yy−1y = \frac{x}{x - 1} \Rightarrow y(x - 1) = x \Rightarrow yx - y = x \Rightarrow x(y - 1) = y \Rightarrow x = \frac{y}{y - 1}. So f−1(x)=xx−1f^{-1}(x) = \frac{x}{x - 1} — ff is its own inverse.

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