Let $S = \{a, b, c\}$ and $T = \{1, 2, 3\}$. Find $F^{-1}$ of the following function $F$ from $S$ to $T$,if it exists: $F = \{(a, 3), (b, 2), (c, 1)\}$.

  • A
    $F^{-1} = \{(3, a), (2, b), (1, c)\}$
  • B
    $F^{-1} = \{(1, a), (2, b), (3, c)\}$
  • C
    $F^{-1} = \{(a, 1), (b, 2), (c, 3)\}$
  • D
    $F^{-1}$ does not exist.

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Let $f(x) = (x - 1)^2 + 1$ for $x \ge 1$.
Statement-$1$: $S = \{x : f(x) = f^{-1}(x)\} = \{1, 2\}$.
Statement-$2$: $f$ is a bijection and $f^{-1}(x) = 1 + \sqrt{x - 1}$ for $x \ge 1$.

State with reason whether the following function has an inverse: $f: \{1,2,3,4\} \rightarrow \{10\}$ with $f = \{(1,10), (2,10), (3,10), (4,10)\}$.

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