Question

Ajutor. Dau coroana
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Answer 1

Explicație pas cu pas:

a)

$E(x) = \bigg(\dfrac{2}{x - 2} + \dfrac{x}{x + 2}\bigg) : \dfrac{x^{2} + 4}{x^{2} - x - 2} =$

$= \dfrac{2(x + 2) + x(x - 2)}{(x - 2)(x + 2)} \cdot \dfrac{x^{2} - x - 2}{x^{2} + 4}$

$= \dfrac{2x + 4 + {x}^{2} - 2x}{(x - 2)(x + 2)} \cdot \dfrac{x^{2} + x - 2x - 2}{x^{2} + 4}$

$= \dfrac{{x}^{2} + 4}{(x - 2)(x + 2)} \cdot \dfrac{(x + 1)(x - 2)}{x^{2} + 4}$

$\bf = \dfrac{x + 1}{x + 2}$

b)

$E(a) \in \mathbb{Z}$

$E(a) = \dfrac{a + 1}{a + 2} = \dfrac{a + 2 - 1}{a + 2} = 1 - \dfrac{1}{a + 2} \in \mathbb{Z} \\ \implies \dfrac{1}{a + 2} \in \mathbb{Z} \iff (a + 2) \in \mathcal{M}_{1}$

$(a + 2) \in \Big\{ - 1; 1\Big\} \implies a \in \Big\{ - 3; - 1\Big\}$