Zadanie 1
[latex]frac{x^2+2x+1}{x^2-1}=frac{(x+1)^2}{(x-1)(x+1)}=frac{x+1}{x-1}[/latex]
======================
Zadanie 2
Wykres (rysunek w załączniku)
[latex]f(x)=frac{2x-3}{x-2}=frac{1+2x-4}{x-2}=frac{1+2(x-2)}{x-2}=[/latex]
[latex]frac{1}{x-2}+frac{2(x-2)}{x-2}=frac{1}{x-2}+2[/latex]
(asymptota pionowa [latex]x=2[/latex], asymptota pozioma [latex]y=2[/latex])
Dziedzina:
[latex]x-2
eq 0[/latex]
[latex]x
eq 2[/latex]
[latex]D=R setminus left{2
ight}[/latex]
Zbiór wartości:
[latex]ZW=R setminus left{2
ight}[/latex]
Miejsce zerowe:
[latex]frac{2x-3}{x-2}=0[/latex]
[latex]2x-3=0[/latex]
[latex]2x=3 /:2[/latex]
[latex]x=1,5[/latex]
Przedziały, w których funkcja przyjmuje wartości dodatnie:
[latex]frac{2x-3}{x-2}>0[/latex]
[latex](2x-3)(x-2)>0[/latex]
[latex]2(x-1,5)(x-2)>0[/latex]
Ramiona paraboli zwrócone do góry. (rysunek w załączniku)
[latex]xin left(- infty;1,5
ight)cup left(2;+ infty
ight)[/latex]
======================
Zadanie 3
[latex]frac{x+5}{x-4}+frac{3}{x}=frac{36}{x^2-4x}[/latex]
[latex]frac{x+5}{x-4}+frac{3}{x}=frac{36}{x(x-4)}[/latex]
Dziedzina:
[latex]x
eq 4[/latex]
[latex]x
eq 0[/latex]
[latex]xin R setminus left{0;4
ight}[/latex]
[latex]frac{x+5}{x-4}+frac{3}{x}-frac{36}{x(x-4)}=0[/latex]
[latex]frac{x(x+5)}{x(x-4)}+frac{3(x-4)}{x(x-4)}-frac{36}{x(x-4)}=0[/latex]
[latex]frac{x(x+5)+3(x-4)-36}{x(x-4)}=0[/latex]
[latex]frac{x^2+5x+3x-12-36}{x(x-4)}=0[/latex]
[latex]frac{x^2+8x-48}{x(x-4)}=0[/latex]
[latex]x^2+8x-48=0[/latex]
[latex]Delta=8^2-4cdot 1cdot (-48)=64+192=256[/latex]
[latex]sqrt{Delta}=sqrt{256}=16[/latex]
[latex]x_1= frac{-8-16}{2}= frac{-24}{2} =-12[/latex]
[latex]x_2= frac{-8+16}{2}= frac{8}{2} =4
ot in D[/latex]
======================
Zadanie 4
[latex]frac{x^2-4x+4}{x^2-4}=frac{(x-2)^2}{(x-2)(x+2)}=frac{x-2}{x+2}[/latex]
======================
Zadanie 5
[latex]frac{1}{R_c}=frac{1}{R_1}+frac{1}{R_2}[/latex]
[latex]frac{1}{R_1}=frac{1}{R_c}-frac{1}{R_2}[/latex]
[latex]frac{1}{R_1}=frac{R_2}{R_cR_2}-frac{R_c}{R_cR_2}[/latex]
[latex]frac{1}{R_1}=frac{R_2-R_c}{R_cR_2}[/latex]
[latex]R_1= frac{R_cR_2}{R_2-R_c}[/latex]
======================
Zadanie 6
[latex]frac{1}{R_c}=frac{1}{R_1}+frac{1}{R_2}[/latex]
[latex]frac{1}{R_c}=frac{R_2}{R_1R_2}+frac{R_1}{R_1R_2}[/latex]
[latex]frac{1}{R_c}=frac{R_2+R_1}{R_1R_2}[/latex]
[latex]R_c= frac{R_1R_2}{R_2+R_1}[/latex]
======================
Zadanie 7
[latex]mleft(m-frac{2}{a}
ight)=2[/latex]
[latex]m^2-frac{2m}{a}=2[/latex]
[latex]frac{2m}{a}=m^2-2[/latex]
[latex]a= frac{2m}{m^2-2}[/latex]
======================
Zadanie 8
[latex]F=mg-momega^2R[/latex]
[latex]mg-momega^2R=F[/latex]
[latex]m(g-omega^2R)=F /:(g-omega^2R)[/latex]
[latex]m=frac{F}{g-omega^2R}[/latex]
======================
Zadanie 9
[latex]frac{x}{x^2+2x+1}-frac{1}{4x^2-4}[/latex]
Dziedzina:
1.
[latex]x^2+2x+1
eq 0[/latex]
[latex](x+1)^2
eq 0[/latex]
[latex]x+1
eq 0[/latex]
[latex]x
eq -1[/latex]
2.
[latex]4x^2-4
eq 0[/latex]
[latex]4(x^2-1)
eq 0 /:4[/latex]
[latex]x^2-1
eq 0[/latex]
[latex](x-1)(x+1)
eq 0[/latex]
[latex]x-1
eq 0 i x+1
eq 0[/latex]
[latex]x
eq 1 i x
eq -1[/latex]
Z 1 i 2
[latex]xin R setminus left{-1;1
ight}[/latex]
[latex]frac{x}{x^2+2x+1}-frac{1}{4x^2-4}=[/latex]
[latex]frac{x}{(x+1)^2}-frac{1}{4(x^2-1)}=[/latex]
[latex]frac{x}{(x+1)^2}-frac{1}{4(x-1)(x+1)}=[/latex]
[latex]frac{4x(x-1)}{4(x-1)(x+1)^2}-frac{x+1}{4(x-1)(x+1)^2}=[/latex]
[latex]frac{4x(x-1)-(x+1)}{4(x-1)(x+1)^2}=[/latex]
[latex]frac{4x^2-4x-x-1}{4(x-1)(x+1)^2}=[/latex]
[latex]frac{4x^2-5x-1}{4(x-1)(x+1)^2}[/latex]
======================
[latex]frac{1}{3x^2-6x+3}-frac{2}{x^2-1}[/latex]
Dziedzina:
1.
[latex]3x^2-6x+3
eq 0 /:3[/latex]
[latex]x^2-2x+1
eq 0[/latex]
[latex](x-1)^2
eq 0[/latex]
[latex]x-1
eq 0[/latex]
[latex]x
eq 1[/latex]
2.
[latex]x^2-1
eq 0[/latex]
[latex](x-1)(x+1)
eq 0[/latex]
[latex]x-1
eq 0 i x+1
eq 0[/latex]
[latex]x
eq 1 i x
eq -1[/latex]
Z 1. i 2.
[latex]xin R setminus left{-1;1
ight}[/latex]
[latex]frac{1}{3x^2-6x+3}-frac{2}{x^2-1}=[/latex]
[latex]frac{1}{3(x^2-2x+1}-frac{2}{(x-1)(x+1)}=[/latex]
[latex]frac{1}{3(x-1)^2}-frac{2}{(x-1)(x+1)}=[/latex]
[latex]frac{x+1}{3(x+1)(x-1)^2}-frac{6(x-1)}{3(x+1)(x-1)^2}=[/latex]
[latex]frac{x+1-6(x-1)}{3(x+1)(x-1)^2}=[/latex]
[latex]frac{x+1-6x+6)}{3(x+1)(x-1)^2}=[/latex]
[latex]frac{-5x+7}{3(x+1)(x-1)^2}[/latex]
