If `f(g(x))=4x^(2)-8x and f(x)=x^(2)-4,` then g(x)=

Options

A. `4-x`
B. `x`
C. `2x-2`
D. 4x

Explanation

To find \( g(x) \) given that \( f(g(x)) = 4x^2 - 8x \) and \( f(x) = x^2 - 4 \), we will follow these steps:

### Step 1: Understand the function composition
We know that \( f(g(x)) = 4x^2 - 8x \). We also have \( f(x) = x^2 - 4 \). This means we need to find \( g(x) \) such that when we substitute \( g(x) \) into \( f(x) \), we get \( 4x^2 - 8x \).

### Step 2: Set up the equation
Since \( f(x) = x^2 - 4 \), we can write:
\[ f(g(x)) = g(x)^2 - 4 \]
We need to set this equal to the expression we have:
\[ g(x)^2 - 4 = 4x^2 - 8x \]

### Step 3: Rearranging the equation
Now, we can rearrange the equation to isolate \( g(x)^2 \):
\[ g(x)^2 = 4x^2 - 8x + 4 \]

### Step 4: Factor the right-hand side
Notice that the right-hand side is a perfect square trinomial:
\[ g(x)^2 = (2x - 4)^2 \]

### Step 5: Taking the square root
Taking the square root of both sides gives us:
\[ g(x) = 2x - 4 \quad \text{or} \quad g(x) = -(2x - 4) \]
However, since we are looking for a function that is consistent with the form of \( f(g(x)) \), we will consider:
\[ g(x) = 2x - 4 \]

### Final Answer
Thus, the function \( g(x) \) is:
\[ g(x) = 2x - 4 \]