Let `f:(2,oo)to X` be defined by f(x)=`4x-x^(2)`. Then f is invertible, if X=

Options

A. `[2,oo]`
B. `(-oo,2]`
C. `(-oo,4)`
D. `[4,oo)`

Explanation

To determine the value of the co-domain \( X \) for the function \( f(x) = 4x - x^2 \) defined on the domain \( (2, \infty) \) such that \( f \) is invertible, we need to analyze the function's properties, particularly its range.

### Step-by-Step Solution:

1. **Identify the Function**:
The function is given by:
\[ f(x) = 4x - x^2 \]

2. **Determine the Nature of the Function**:
This is a quadratic function, which can be rewritten in standard form:
\[ f(x) = -x^2 + 4x \]
Here, \( a = -1 \) (which indicates the parabola opens downwards), \( b = 4 \), and \( c = 0 \).

3. **Find the Vertex**:
The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ x = -\frac{4}{2(-1)} = 2 \]
To find the corresponding \( y \)-coordinate of the vertex, we substitute \( x = 2 \) back into the function:
\[ f(2) = 4(2) - (2^2) = 8 - 4 = 4 \]
Thus, the vertex is at the point \( (2, 4) \).

4. **Determine the Roots**:
To find the roots of the function, set \( f(x) = 0 \):
\[ 4x - x^2 = 0 \implies x(4 - x) = 0 \]
This gives us the roots \( x = 0 \) and \( x = 4 \).

5. **Analyze the Function on the Domain**:
The function is defined on the interval \( (2, \infty) \). Since the vertex \( (2, 4) \) is the maximum point of the parabola and the function decreases as \( x \) increases beyond 2, we can conclude:
- The function will take values from \( 4 \) down to \( -\infty \) as \( x \) approaches \( \infty \).

6. **Determine the Co-domain \( X \)**:
Since \( f \) is defined from \( (2, \infty) \) and we found that the maximum value is \( 4 \) (not included since the domain does not include \( 2 \)), the range of \( f \) is:
\[ (-\infty, 4) \]
Therefore, the co-domain \( X \) is:
\[ X = (-\infty, 4) \]

### Final Answer:
Thus, the value of \( X \) is:
\[ X = (-\infty, 4) \]