Odpowiedź:
zad 2
P = (3√2 , - √2) ; f(x)= a/x dla x ≠ 0
- √2 = a/3√2
a = - √2 * 3√2 = - 3 * 2 = - 6
f(x) = -6/x ; Q = ( -3 , 2 )
2 = - 6/(-3)
2= 6/3
2 = 2
L = P
Punkt Q należy do wykresu funkcji
zad 4
a)
(x - 3)/x = (x+1)/(x - 8) dla x ≠ 0 i x ≠ 8
(x - 3)/x = (x + 1)/(x - 8)
(x - 3)(x - 8)= x(x + 1)
x² - 3x - 8x + 24 = x² + x
x² - 11x + 24 = x² - x
x² - x² - 11x + x = - 24
- 10x = - 24
10x = 24
x = 24/10 = 2,4
b)
(x² + 2x - 3)/(x + 3) = 2x dla x ≠ - 3
(x² + 2x - 3)/(x + 3) = 2x
x² + 2x - 3 = 2x(x + 3)
x² + 2x - 3 = 2x² + 6x
x²-2x² + 2x - 6x - 3 = 0
- x² - 4x - 3 = 0
a = - 1 , b = - 4 , c = - 3
Δ = b² - 4ac = (- 4)² - 4 *(- 1) * ( - 3) = 16 - 12 = 4
√Δ = √4 = 2
x₁ = ( - b - √Δ)/2a = (4 - 2)/(- 2) = 2/(- 2) = - 2/2 = - 1
x₂ = (- b + √Δ)/2a = (4 + 2)/(- 2)= 6/(- 2) = - 6/2 = - 3
zad 5
a)
(4x-12)/2x² : (3x - 9)/8x = 4(x - 3)/2x² : 3(x - 3)/8x dla x ≠ 0 i x ≠ 3
4(x - 3)/2x² : 3(x - 3)/8x = 4(x - 3)/2x² * 8x/3(x -3) = 4/x * 4/3 = 16/(3x)
b)
(x + 3)/(x - 2) - ( x - 1)/(x + 2) dla x ≠ 2 i x ≠ - 2
(x + 3)/(x - 2) - ( x - 1)/(x + 2) = [(x + 3)(x + 2) - (x - 1)(x - 2)]/[(x - 2)(x + 2)]=
= [x² + 3x + 2x + 6 - (x² -x - 2x + 2)]/(x² - 4) =
= [x² + 5x + 6 - (x² - 3x + 2)]/(x² - 4) = (x² + 5x + 6 - x² + 3x - 2)/(x² - 4) =
= (8x + 4)/(x² - 4) = 4(2x + 1)/(x² - 4)
