If `f (x) =3 -x and g (x) = (x ^(2))/(2),` which of the following is NOT in the range of `f (g (x)` ?

Options

A. `-3`
B. 0
C. 2
D. 4

Explanation

To solve the problem, we need to find the range of the composite function \( f(g(x)) \) where \( f(x) = 3 - x \) and \( g(x) = \frac{x^2}{2} \).

### Step 1: Find \( g(x) \)
The function \( g(x) \) is given by:
\[ g(x) = \frac{x^2}{2} \]
This function takes any real number \( x \) and outputs a non-negative value since \( x^2 \) is always non-negative. Therefore, the range of \( g(x) \) is:
\[ [0, \infty) \]

### Step 2: Substitute \( g(x) \) into \( f(x) \)
Next, we substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f\left(\frac{x^2}{2}\right) = 3 - \frac{x^2}{2} \]

### Step 3: Determine the range of \( f(g(x)) \)
Now, we need to analyze the function \( f(g(x)) = 3 - \frac{x^2}{2} \).

1. The term \( \frac{x^2}{2} \) takes values from \( 0 \) to \( \infty \) as \( x \) varies over all real numbers.
2. The maximum value of \( f(g(x)) \) occurs when \( \frac{x^2}{2} \) is at its minimum, which is \( 0 \):
\[ f(g(x)) = 3 - 0 = 3 \]
3. As \( x^2 \) increases, \( \frac{x^2}{2} \) increases, causing \( f(g(x)) \) to decrease. Therefore, as \( x^2 \) approaches infinity, \( f(g(x)) \) approaches negative infinity:
\[ \lim_{x \to \infty} f(g(x)) = 3 - \infty = -\infty \]

Thus, the range of \( f(g(x)) \) is:
\[ (-\infty, 3] \]

### Step 4: Identify the value NOT in the range
From the range \( (-\infty, 3] \), we can see that any number greater than \( 3 \) is not in the range. For example, \( 4 \) is greater than \( 3 \) and therefore is not in the range of \( f(g(x)) \).

### Conclusion
The value that is NOT in the range of \( f(g(x)) \) is:
\[ \boxed{4} \]