Consider the functions f and g where f (x) = 3x − 5 and g (x) = x − 2 . (a) Find the inverse function, f^−1 . (b) Given that 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) Start with f(x)= 3x − 5; y=3x - 5, and switch x and y before rearanging to get y in terms of x again:

y=3x - 5

x=3y - 5

(x+5)/3=y

Therefore f^−1(x)=(x+5)/3

(b) Start with f(x)= 3x − 5 and and sub result into g^−1(x) = x + 2:

(g^−1 o f)(x) = (3x − 5) + 2 = 3x - 3

(c) (f^−1 o g)(x) = (g^−1 o f)(x)

(x + 3)/3 = 3x - 3

x + 3 = 9x - 9

12 = 8x

3/2 = x