For each pair of functions, find f(g(x)) and g(f(x)). Simplify your answers.

Question

f(x)= 1/x+5, g(x)=2/x-3

Answer 1

depends on how you intended any parentheses

as written, it's g(x) = 2/x - 3 = (2-3x).x

and f(g(x)) = 2/(2-3x) + (10-15x)/(2-3x) = (10-14x)/(2-3x)

similar ambiguity in the 2nd problem

Answer 2

f(g(x)):

2/(x-3) + 5 = 2/(x-3) + 5(x-3)/(x-3)

= [2 + 5(x-3) ] /(x-3)

= [2 + 5x-15]/(x-3)

(5x-13)/(x-3)

the reciprocal is (x-3)/(5x-13) = f(g(x))

----------------------------------------------------

g(f(x))

1/(x+5) - 3 = 1/(x+5) - (x+5)/(x+5)

= [1 - (x+5)]/(x+5)

= [ 1 -x -5)/(x+5)

= (-x-4)/(x+5)

= -(x+4)/(x+5)

so g(f(x)) = -2(x+4)/(x+5)