If `f(x)=2x^(3)` and `g(x)=3x`, what is the value of `g(f(-2))-f(g(-2))` ?

Options

A. `-480`
B. `-384`
C. 0
D. 384

Explanation

To solve the problem, we need to find the value of \( g(f(-2)) - f(g(-2)) \) given the functions \( f(x) = 2x^3 \) and \( g(x) = 3x \).

### Step 1: Calculate \( f(-2) \)

Using the function \( f(x) = 2x^3 \):

\[ f(-2) = 2(-2)^3 \]

Calculating \( (-2)^3 \):

\[ (-2)^3 = -8 \]

Now substituting back:

\[ f(-2) = 2 \times -8 = -16 \]

### Step 2: Calculate \( g(f(-2)) \)

Now we need to find \( g(f(-2)) \), which is \( g(-16) \):

Using the function \( g(x) = 3x \):

\[ g(-16) = 3 \times -16 \]

Calculating this:

\[ g(-16) = -48 \]

### Step 3: Calculate \( g(-2) \)

Next, we calculate \( g(-2) \):

\[ g(-2) = 3 \times -2 \]

Calculating this:

\[ g(-2) = -6 \]

### Step 4: Calculate \( f(g(-2)) \)

Now we need to find \( f(g(-2)) \), which is \( f(-6) \):

Using the function \( f(x) = 2x^3 \):

\[ f(-6) = 2(-6)^3 \]

Calculating \( (-6)^3 \):

\[ (-6)^3 = -216 \]

Now substituting back:

\[ f(-6) = 2 \times -216 = -432 \]

### Step 5: Calculate \( g(f(-2)) - f(g(-2)) \)

Now we can find the final result:

\[ g(f(-2)) - f(g(-2)) = -48 - (-432) \]

This simplifies to:

\[ -48 + 432 = 384 \]

Thus, the final answer is:

\[ \boxed{384} \]