Question

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

f(x) = quantity x minus nine divided by quantity x plus five. and g(x) = quantity negative five x minus nine divided by quantity x minus one.
Show work pls

Answer 1

See below

Step-by-step explanation

Given

`f(x) = \frac{ {x}- 9 }{x + 5}`

and

`g(x) = ( - 5x - 9)/(x - 1)`

`(f \circ \: g)(x)= ( (( - 5x - 9)/(x - 1))- 9 )/((( - 5x - 9)/(x - 1) )+ 5)`

`(f \circ \: g)(x)= ( ( - 5x - 9 - 9(x - 1))/(x - 1))/(( - 5x - 9 + 5(x - 1))/(x - 1) )`

Expand:

`(f \circ \: g)(x)= ( ( - 5x - 9 - 9x + 9)/(x - 1))/(( - 5x - 9 + 5x - 5)/(x - 1) )`

`(f \circ \: g)(x)= ( ( - 5x - 9x + 9 - 9)/(x - 1))/(( - 5x + 5x - 5 - 9)/(x - 1) )`

`(f \circ \: g)(x)= ( ( - 14x )/(x - 1))/(( -14)/(x - 1) )`

Since the denominators are the same, they will cancel out,

`(f \circ \: g)(x)= ( - 14x)/( - 14) = x`