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

Answer 1

Although no brackets, I think these are the functions:\[f(x)=\frac{ x-9 }{ x+5 }\]and\[g(x)=\frac{ -5x-9 }{ x-1 }\]
If we have to calculate f(g(x)), we must take the outcome of function g and put that into function f, so we get:\[f(g(x))=f \left( \frac{ -5x-9 }{ x-1 } \right)\]So with this weird thing, we need to do all the calculations of f!

Answer 2

Here I go:\[f \left( \frac{ -5x-9 }{ x-1 } \right)=\frac{ \frac{ -5x-9 }{ x-1 }-9 }{ \frac{ -5x-9 }{ x-1 } +5}\]Now all we have to do is juggle the numbers in such a way that only one x is left...
So we multiply numerator and denominator with the same number x-1 to get rid of the fractions-in-fractions:\[\frac{ -5x-9-9(x-1) }{ -5x-9+5 \cdot(x-1) }\]Now work out the brackets and simplitfy as much as we can:\[\frac{ -5x-9-9x+9 }{ -5x-9+5x-5 }=\frac{ -14x }{ -14 }=x\]So that one's a success!

Why not try g(f(x)) yourself now?
Just ask if you need help with it!