# 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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