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ISI2021-MMA: 27

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Consider two real valued functions $f$ and $g$ given by $$f(x)=\frac{x}{x-1} \; \text{for } x>1, \quad \text{and }\quad g(x)=7-x^{3} \; \text{for } x \in \mathbb{R}.$$ Which of the following statements about inverse functions is true?

  1. Neither $f^{-1}$ nor $g^{-1}$ exists
  2. $f^{-1}$ exists, but not $g^{-1}$
  3. $f^{-1}$ does not exist, but $g^{-1}$ does
  4. Both $f^{-1}$ and $g^{-1}$ exist.
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