\(A=\dfrac{f\left(x_1\right)-f\left(x_2\right)}{x_1-x_2}=\left(\dfrac{x_1+1}{x_1-2}-\dfrac{x_2+1}{x_2-2}\right):\left(x_1-x_2\right)\)
\(=\dfrac{x_1x_2-2x_1+x_2-2-x_1x_2+2x_2-x_1+2}{\left(x_2-2\right)\left(x_1-2\right)}\cdot\dfrac{1}{x_1-x_2}\)
\(=\dfrac{-3x_1+3x_2}{\left(x_2-2\right)\left(x_1-2\right)}\cdot\dfrac{1}{x_1-x_2}=\dfrac{-3}{\left(x_2-2\right)\left(x_1-2\right)}\)
Trường hợp 1: x<2
=>\(\left(x_1-2\right)\left(x_2-2\right)>0\)
=>A<0
=>Hàm số nghịch biến
Trường hợp 2: x>2
=>\(\left(x_1-2\right)\left(x_2-2\right)>0\)
=>A<0
=>Hàm số nghịch biến