trig problem?

Question

i have to show that g(f(x)) = x and (f(g(x)) = x.

f(x) = 2 / (1-x)
g(x) = (x-2) / x

i tried doing it but its not working out. can someone please show me how to do it step by step?

Answer 1

f [ g ( x ) ]
f [ ( x - 2 ) / x ]
f (y)
2 / ( 1 - y )
2 / [1 - (x - 2) / x ]
2 x / [ x - x + 2 ]
2 x / 2
x

g ( f ( x ) )
g [ 2 / (1 - x) ]
g ( y )
( y - 2 ) / y
[ 2 / ( 1 - x ) - 2 ] / [ 2 / ( 1 - x ) ]
1 - ( 1 - x )
x

Answer 2

f(g(x)):
2/(1-(x-2)/x)

make 1=x/x:
2/(x/x-(x-2)/x)

simplify:
2/((x-x+2)/x

x's cancel, flip and multiply:
2*(x/2)

we get x

g(f(x)) will be similar, but a little messier
try doing it yourself and let me know if you need more help

common denominators FTW