Question

Fie expresia E(x)=(x+1/x-1+ x-1/x+1+ 1-x+2xpatrat/1-xpatrat):xpatrat+2x+1/x la a 3a -x
A stabitilti domeniul maxim de existenta al expresesiei
B Aratati ca e(x)=x/x+1
C rezolvati ecuatia 1/E(x)=2

Answer 1

Domeniul maxim de existenta:
O fractie nu are sens daca numitorul este 0. Asa ca vom cauta x a.i numitorul sa fie 0
$x - 1 = 0 = > x = 1 \\ \\ x + 1 = 0 = > x = - 1 \\ \\ 1 - {x}^{2} = 0 \\ {x}^{2} = 1 \\ |x| = 1 \\ = > x= 1 \\ = > x = - 1 \\ \\ {x}^{3} - x = 0 \\ x( {x}^{2} - 1) = 0 \\ = > x = 0 \\ = > {x}^{2} - 1 = 0$
Deci x apartine multimii R minus { -1, 1 , 0}
B.
$( \frac{x + 1}{x - 1} + \frac{x - 1}{x + 1} + \frac{1 - x + 2 {x}^{2} }{ - (x - 1)( x+ 1)} ) \times \frac{ {x}^{3} - x}{ {x}^{2} + 2x + 1}$
Prima paranteza o scrii cu acelasi numitor.
$( \frac{(x + 1)( x+ 1)}{(x + 1)(x - 1)} + \frac{(x - 1)(x - 1)}{(x - 1)(x + 1)} - \frac{1 - x + 2 {x}^{2} }{(x - 1)(x + 1)} ) \times \frac{x(x - 1)(x + 1)}{(x + 1)^{2} } \\ \\ \frac{ {(x + 1)}^{2} + {(x - 1)}^{2} - (1 -x + 2 {x}^{2} }{(x - 1)(x + 1)} \times \frac{x(x - 1)(x + 1)}{ {(x + 1)}^{2} }$
Simplifici x - 1 si x + 1 si inmultesti ce a ramas

$( {(x + 1)}^{2} + {(x - 1)}^{2} - 1 + x - 2 {x}^{2} ) \times \frac{x}{ {(x + 1)}^{2} } \\ \frac{x( {(x + 1)}^{2} + {(x - 1)}^{2} - 1 + x - 2 {x}^{2} )}{ {(x + 1)}^{2} }$
Desfaci acele binom-uri

$\frac{x( {x}^{2} + 2x + 1 + {x}^{2} - 2x + 1 - 1 + x - 2 {x}^{2}) }{ {(x + 1)}^{2} } \\ \frac{x(1 + x)}{ {(x + 1)}^{2} } \\ \frac{x(x + 1)}{ {(x + 1)}^{2} } \\ \frac{x}{x + 1}$
C.
$\frac{1}{ \frac{x}{x + 1} } = 2 \\ \frac{x + 1}{x} = 2 \\ 2x = x + 1 \\ 2x - x = 1 \\ x = 1$