Question

Sa se arate ca urmatoarea functie este bijectiva si sa se afle inversa acesteia:
A) f:R/{-1}-R/{3}, f(x)=6X+1/X+2

Answer 1

Fie x1,x2 ∈ R/{-1} a.i f(x1)=f(x2)
$\frac{6*x1+1}{x1+2}= \frac{6*x2+1}{x2+2}$
⇔ (6*x1+1)(x2+2)=(6*x2+1)(x1+2)
⇔ 6*x1*x2+12*x1+x2+2=6*x1*x2+12*x2+x1+2  | - 6*x1*x2 - 2
⇔ 12*x1+x2=12*x2+x1⇔11*x1=11*x2⇔x1=x2 (A) ⇒ f. injectiva (1)
Fie y∈ R/{3} a.i f(x)=y
$\frac{6x+1}{x+2}=y$
⇔ 6x+1=xy+2y⇔6x-xy=2y-1⇔x(6-y)=2y-1⇔x=$\frac{2y-1}{6-y}$∈R/{3} ⇒f.surjectiva (2)
(1)si(2)⇒ f.bijectiva
$f ^{-1}(x)= \frac{2x-1}{6-x}$