Given two functions f and g where f(x)=3x-5 and g(x)=x-2. Find: a) the inverse f^-1(x), b) given g^-1(x)=x+2, find (g^-1 o f)(x), c) given also that (f^-1 o g)(x)=(x+3)/3, solve (f^-1 o g)(x)=(g^-1 o f)(x)

a)  For an inverse function- "inputs become outputs" so swap the positions of the input-variable (i.e "x") with the output variable (i.e f(x)) and then rearange. Once rearanged so that f(x) is on the left and any other variables are on the right, the correct notation to use is now f^-1(x) (rather than f(x)) to signify that this is now an inverse. 

x=3f(x)-5

x+5=3f(x)

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

b) The key with this question is to understand the notation because the math is quiet simple. 

"(g^-1 o f)(x)" is sort of like a command. It is telling you to take f(x) as your new input (i.e your "x" variable) and put it into g^-1(x) to get a new output (i.e. g^-1 o f)(x)) ... Think of it like a command telling you to do this: 

where we know, f(x)=3x-5 and g^-1(x)=x+2

(g^-1 o f)(x)= f(x) +2

therefore, 

(g^-1 o f)(x)= (3x-5)+2

(g^-1 o f)(x)= 3x-3

c) Just a case of plugging in what you see and following the directions of the question

(f^-1 o g)(x)=(g^-1 o f)(x)

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

x+3=9x-9

8x=12

x=3/2