Question

Fie expresia : E(x) =(X+1/x - X-1/x+1 +1-x/Xpatrat +x):2x-4/Xpatrat+3x

a)Stabilti domeniul maxim de exisetenta al expresiei
b)Aratati ca E(x)=x+3/x-2
c)determinati valorile intregi ale lui x pentru care E(X) apartine Z
Dau coroana

Answer 1

a)Numitorul unei fractii nu poate fii zero, deci cautam x a.I numitorul sa fie 0
x = 0
x + 1 = 0 => x = -1
${x}^{2} + x = 0 \\ x(x + 1) = 0 \\ = > x = 0 \\ = > x + 1 = 0 = > x = - 1$
${x}^{2} + 3x = 0 \\ x(x + 3) = 0 \\ = > x = 0 \\ = >x + 3 = 0 = > x = - 3$
=> x apartine lui R \ { -3 , -1 , 0}

b)

$( \frac{x + 1}{x} - \frac{x - 1}{x + 1} + \frac{1 - x}{x(x + 1)}) \times \frac{ {x}^{2} + 3x }{2(x - 2)} \\ ( \frac{x + 1}{x} + \frac{ - x(x - 1) + 1 - x}{x(x + 1)} ) \times \frac{x(x + 3)}{2(x - 2)} \\ ( \frac{x + 1}{x} + \frac{ - {x}^{2} + x + 1 - x}{x(x + 1)} ) \times \frac{x(x + 3)}{2(x - 2)} \\ (\frac{x + 1}{x} + \frac{ 1 - {x}^{2} }{x(x + 1)} ) \times \frac{x(x + 3)}{2(x - 2)} \\ ( \frac{x + 1}{x} + \frac{(1 - x)(1 + x)}{x(x + 1)} ) \times \frac{x(x + 3)}{2(x - 2)} \\ ( \frac{x + 1}{x} + \frac{1 - x}{x} ) \times \frac{x(x + 3)}{2(x - 2)} \\ \frac{x + 1 + 1 - x}{x} \times \frac{x(x + 3)}{2(x - 2)} \\ 2 \times \frac{x + 3}{2(x - 2)} \\ \frac{x + 3}{x - 2}$
c)
Fractia aceasta apartine lui Z

$x - 2 \: | \: x + 3 \\ x - 2 |x - 2$
Partile din dreapta le scazi
=>
$x - 2 |5$
x - 2 apartine { -5, -1, 1, 5}
x apartine {-3; 3 ; 7}

( 2 nu poate fi pt ca numitorul ar fi 0)