Suppose the graph of `f(x)=-x^(3)` is translated 4 units right and 2 units down, resulting in the graph of a new function g. what is the value of g(-2)?

Options

A. `-218`
B. `-10`
C. 6
D. 214

Explanation

To solve the problem, we need to follow these steps:

### Step 1: Understand the Translation
The function \( f(x) = -x^3 \) is translated 4 units to the right and 2 units down. This means we need to adjust the function accordingly.

### Step 2: Apply the Translation
1. **Translation 4 units to the right**: To translate a function \( f(x) \) to the right by \( a \) units, we replace \( x \) with \( x - a \). Here, \( a = 4 \), so we replace \( x \) with \( x - 4 \).
\[ f(x) \rightarrow f(x - 4) = - (x - 4)^3 \]

2. **Translation 2 units down**: To translate a function down by \( b \) units, we subtract \( b \) from the function. Here, \( b = 2 \), so we subtract 2 from the function:
\[ g(x) = f(x - 4) - 2 = - (x - 4)^3 - 2 \]

### Step 3: Write the New Function
Thus, the new function \( g(x) \) after the translations is:
\[ g(x) = - (x - 4)^3 - 2 \]

### Step 4: Find \( g(-2) \)
Now we need to find the value of \( g(-2) \):
\[ g(-2) = -((-2) - 4)^3 - 2 \]
Calculating the expression inside the parentheses:
\[ g(-2) = -(-6)^3 - 2 \]
Calculating \( (-6)^3 \):
\[ (-6)^3 = -216 \]
Now substituting back into the equation:
\[ g(-2) = -(-216) - 2 = 216 - 2 = 214 \]

### Final Answer
Thus, the value of \( g(-2) \) is:
\[ \boxed{214} \]