Piecewise Functions - 4

Algebra Level 5

Let a function \(f : \mathbb{R} - [0, 1) \rightarrow \mathbb{R}^+\) be defined as

\[f(x) = \begin{cases} g(x) & \text{if }x \in \mathbb{Z}^+\\ h(x) & \text{if } x \in \mathbb{Z}^- \\ x\lfloor x \rfloor & \text{otherwise} \end{cases} \]

If \(f^{-1}: \mathbb{R}^+ \mapsto \mathbb{R} - [0, 1)\) is a function, and \(g\) and \(h\) are polynomials of degree \(2\),

Find \(5 \lfloor g(3.5) - h(3.5)\rfloor\).

Details and assumptions

  • \(\mathbb{R} - [0,1)\) is the interval \((-\infty, 0) \cup [1, \infty)\)
  • \(\mathbb{R}^+\) is the set of all positive reals, that is \((0, \infty)\)
  • \(\mathbb{Z}^+\) is the set of all integers \(> 0\)
  • Similarly, \(\mathbb{Z}^-\) is the set of all integers \(< 0\)
  • \(\lfloor \cdot \rfloor\) is the floor function

This problem is part of the set - Piecewise-defined Functions

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