What is the inverse of g(f(x)) if g(x) = 3x^3 - 4 and f(x) = x - 2?

To determine the inverse of the composition ( g(f(x)) ), we first need to understand how the functions ( g(x) ) and ( f(x) ) interact.

Starting with the function definitions, we have:

• ( f(x) = x - 2 )

• ( g(x) = 3x^3 - 4 )

The composition ( g(f(x)) ) involves substituting ( f(x) ) into ( g(x) ). This means we replace ( x ) in ( g(x) ) with ( f(x) ):

[

g(f(x)) = g(x - 2) = 3(x - 2)^3 - 4

]

Next, we simplify ( g(f(x)) ):

1. First, calculate ( (x - 2)^3 ):

[

(x - 2)^3 = x^3 - 6x^2 + 12x - 8

]

2. Now substituting this back into ( g(f(x)) ):

[

g(f(x)) = 3(x^3 - 6x^2 + 12x -