We are given two functions:

• $f(x) = 5 - 2x$
• $g(x) = \frac{x^2}{4}$

We are asked to determine which value is not in the range of $f(g(x))$.

Step 1: Find the composition $f(g(x))$

We substitute $g(x)$ into $f(x)$:

$f(g(x)) = 5 - 2 \cdot g(x) = 5 - 2 \left(\frac{x^2}{4}\right) = 5 - \frac{x^2}{2}$

Thus, the composite function is:

$f(g(x)) = 5 - \frac{x^2}{2}$

Step 2: Analyze the range of $f(g(x))$

Now, let's consider the range of this function. Since $g(x) = \frac{x^2}{4}$, $x^2$ is always non-negative, so $g(x) \geq 0$. Therefore, $f(g(x))$ is bounded as follows:

$f(g(x)) = 5 - \frac{x^2}{2}$

As $x^2$ increases, the value of $f(g(x))$ decreases. The minimum value of $f(g(x))$ occurs when $x^2 \to \infty$, making $f(g(x)) \to -\infty$. The maximum value occurs when $x = 0$, which gives:

$f(g(0)) = 5 - \frac{0^2}{2} = 5$

Thus, the range of $f(g(x))$ is $(-\infty, 5]$.

Step 3: Determine which value is not in the range

Looking at the options:

• $-3$ is in the range because it's less than 5.
• $0$ is in the range because it's less than 5.
• $5$ is in the range because it's the maximum value.
• $6$ is not in the range because the range is $(-\infty, 5]$.

Thus, the answer is:

$\boxed{D \ 6}$

Would you like more details on any step? Here are five questions for further exploration:

1. How do you determine the range of a composite function?
2. What would happen to the range if the function $f(x)$ were different?
3. Can $f(g(x))$ ever have an upper bound greater than 5? Why or why not?
4. How do you find the range of more complex compositions of functions?
5. What is the effect of squaring a variable in determining a function's range?

Tip: Always analyze the behavior of the inner function first when finding the range of a composite function!