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

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

• A. cbrt((x + 4)/3) + 2
• B. (x + 4)/3
• C. 3cbrt(x + 4)
• D. cbrt(x) - 2

Answer: (A)

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 -

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 -