Question

ALGEBRA 2!!!!!!!!! SHOW YOUR WORK!!!!!!!!!!!!!
Do f(g(x)) and g(f(x))

Answer 1

`\bf f(x)=\cfrac{2x-3}{x+1}~\hspace{10em}g(x)=\cfrac{x+3}{2-x} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(~~g(x)~~)\implies \cfrac{2[g(x)]-3}{[g(x)]+1}\implies \cfrac{2\left( (x+3)/(2-x) \right)-3}{\left( (x+3)/(2-x) \right)+1}\implies \cfrac{(2x+6)/(2-x)-3}{(x+3)/(2-x)+1} \\\\\\ \cfrac{(2x+6-6+3x)/(2-x)}{(x+3+2-x)/(2-x)}\implies \cfrac{2x+6-6+3x}{2-x}\cdot \cfrac{2-x}{x+3+2-x}\implies \cfrac{5x}{5}\implies x`

`\bf \rule{34em}{0.25pt}\\\\ g(~~f(x)~~)\implies \cfrac{[f(x)]+3}{2-[f(x)]}\implies \cfrac{(2x-3)/(x+1)+3}{2-(2x-3)/(x+1)}\implies \cfrac{(2x-3+3x+3)/(x+1)}{(2x+2-(2x-3))/(x+1)} \\\\\\ \cfrac{2x-3+3x+3}{x+1}\cdot \cfrac{x+1}{2x+2-(2x-3)}\implies \cfrac{2x-3+3x+3}{x+1}\cdot \cfrac{x+1}{2x+2-2x+3} \\\\\\ \cfrac{5x}{5}\implies x`

and in case you recall your inverses, when f( g(x) ) = x, or g( f(x) ) = x, simply means, they're inverse of each other.