Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.
f(x) = x-9/x+5
and g(x) = -5x-9/x-1
PLEASE HELP I GIVE MEDALS

Answer 1

So, remember when you find the inverse of a function you turn the "f(x)" in y, then set that equal to the function, then switch the place of y and x, then solve for y. (get y by itself) Thats how you would get g(x) (assuming we aren't given g(x) like in your problem).

Thus, to check if the given functions are inverses of each other, find the composite of the f(x) "f of x" and g(x) "g of x", which is the same as [f o g], f(g(x)) read "f of (g of x)" (remember g(x) is read as "g of x".)
Now,
\[f(x)=\frac{ x-9 }{ x+5 }\]
and \[g(x)=\frac{ 5x+9 }{ x-1 }\]
\[f(g(x))\] NOTICE: \[g(x)=\frac{ 5x+9 }{ x-1 }\]
So, just replace it.
\[f(\frac{ 5x+9 }{ x-1 })= \frac{ x-9 }{ x+5 }\]
(remember: if f(x)=x+3, find what x=2 is?...
f(2)=2+3
f(2)=5 )
Do the with the eq
\[f(\frac{ 5x+9 }{ x+1 })=\frac{(\frac{ -5x-9 }{ x+1 })-9 }{ (\frac{ -5x-9 }{ x+1 })+5 }\]
Then, solve, if f(g(x))=x then, they are inverse.
Though, it would be a lot easier to just find the inverse of f(x), and compare it to g(x)