Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

Answer 1

f(x)=
`(x-7)/(x+3)`
g(x)=
`(-3x-7)/(x-1)`

1. f(g(x))=x
f(g(x)) means to put g(x) wherever you find x in f(x).

f(g(x)) =
`( (-3x-7)/(x-1)-7 )/( (-3x-7)/(x-1) +3)`
f(g(x)) =
`( (-3x-7-7(x-1))/(x-1) )/( (-3x-7+3(x-1))/(x-1) )`
Reverse the dominator and simplify with x-1

f(g(x)) =
`(-3x-7-7x+7)/(-3x-7+3x-3)`
f(g(x)) =
`(-10x)/(-10)` = x

So we proved that f(g(x))=x

2. g(f(x))=x
g(f(x)) means to put f(x) wherever you find x in g(x).

g(f(x)) =
`(-3( (x-7)/(x+3))-7)/( (x-7)/(x+3)-1 )`
g(f(x)) =
`( (-3x+21)/(x+3) -7)/( (x-7-x-3)/(x+3) )`
g(f(x)) =
`( (-3x+21-7x-21)/(x+3) )/( (x-7-x-3)/(x+3) )`

Reverse the dominator and simplify with x+3

g(f(x))=
`(-10x)/(-10) =x`

So we proved that g(f(x))=x

Because f(g(x)) = x and g(f(x)) = x f and g are inverse.