A function dissimiliarity

Algebra Level 4

Find the number of integral values of $x$ satisfying the equation $\text{sgn} \left( \left \lfloor \dfrac{15}{1+x^2} \right \rfloor \right) = \lfloor 1 + \{2x \} \rfloor$

Notations:

$\lfloor \cdot \rfloor$ denotes the floor function .
$\{ \cdot \}$ denotes the fractional part function .
$\text{sgn} (\cdot)$ denotes the signum function , if $\text{sgn}(x) = \begin{cases} -1, \text{ if } x < 0 \\ 0, \text{ if } x = 0 \\ 1, \text{ if } x > 0 \end{cases}$ .

5 7 15 16

1 solution

X X
Jun 28, 2020

Since $x$ is an integer, $\{2x\}=0$ , hence $\lfloor 1+\{2x\}\rfloor=1=\text{sgn}\left(\lfloor{\dfrac{15}{1+x^2}}\rfloor\right)$

This means $\lfloor{\dfrac{15}{1+x^2}}\rfloor>0$ , which is equivalent to $\dfrac{15}{1+x^2}\geq1$

$14\geq x^2$ , so there are 7 possible integers (from -3 to 3) satisfying the condition.

A function dissimiliarity

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