Question

Se considera functia f:R\{-1}->R, f(x)=(-x^2-2x-3)÷(x^2+2x+1).Determinati Imf

Answer 1

[tex]Notam\ f(x)=y\\ \frac{-x^2-2x-3}{x^2+2x+1}=y\\ -x^2-2x-3=yx^2+2yx+y\\ x^2(-y-1)-2(y+1)x-3-y=0\\ \Delta=4(y+1)^2-4(-y-1)(-3-y)\\ \Delta=4y^2+8y-4(3y+3+y^2+y)\\ \Delta=4y^2+8y-12y+12-4y^2-4y\\ \Delta=-8y+12\\ \Delta\geq0\Rightarrow-8y+12\geq 0\\ ~~~~~~~~~~~~~~12\geq 8y\\ ~~~~~~~~~~~~~~y\leq \frac{3}{2}\Rightarrow y\in (-\infty,\frac{3}{2}]\\ Asadar:imf=(-\infty,\frac{3}{2}] [/tex]