Bonjour,
1)
[tex] 2x-3+\dfrac{1}{x+1} \\
=\dfrac{(2x-3)(x+1)+1}{x+1} \\
=\dfrac{2x^2-3x+2x-3+1}{x+1} \\
=\dfrac{2x^2-x-2}{x+1} \\
[/tex]
2)
[tex] f(\dfrac{2}{3}) =2*\dfrac{2}{3}-3+\dfrac{1}{\dfrac{2}{3}+1}\\
=\dfrac{4}{3}-\dfrac{9}{3}+\dfrac{3}{2+3}\\
=-\dfrac{5}{3}+\dfrac{3}{5}\\
=\dfrac{-25+9}{15}\\
=\dfrac{-16}{15}\\
\approx{-1,06666666...}
[/tex]
[tex] f(\sqrt{5}) =2\sqrt{5} -3+\dfrac{1}{\sqrt{5}+1} \\
=2\sqrt{5} -3+\dfrac{\sqrt{5}-1 }{5-1} \\
=\dfrac{9}{4}\sqrt{5}- \dfrac{13}{4} \\
=\dfrac{1}{4}(9\sqrt{5}- 13) \\
\approx{1.781152...
[/tex]
[tex] f(1+\sqrt{3})=2( 1+\sqrt{3})-3+\dfrac{1}{1+\sqrt{3}+1} \\
=- 1+2\sqrt{3}+2-\sqrt{3} \\
=1+\sqrt{3} \\
\approx{2.73205080...} [/tex]
