Question

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

Answer 1

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

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

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.