To solve the problem, we need to find the range of $f(g(x))$ and determine which of the options cannot be in this range.

Given functions:

1. $f(x) = -3 - 2x$
2. $g(x) = -\frac{x^2}{6}$

Step 1: Determine $f(g(x))$

Substitute $g(x)$ into $f(x)$: $f(g(x)) = f\left(-\frac{x^2}{6}\right)$ Using the definition of $f(x) = -3 - 2x$, replace $x$ in $f(x)$ with $-\frac{x^2}{6}$: $f\left(-\frac{x^2}{6}\right) = -3 - 2\left(-\frac{x^2}{6}\right)$ $= -3 + \frac{x^2}{3}$

Step 2: Determine the Range of $f(g(x))$

Since $f(g(x)) = -3 + \frac{x^2}{3}$, analyze $\frac{x^2}{3}$:

• $x^2$ is always non-negative, so $\frac{x^2}{3} \geq 0$.
• Therefore, $-3 + \frac{x^2}{3} \geq -3$.

The range of $f(g(x))$ is $[-3, \infty)$.

Step 3: Determine Which Option is Not in the Range

Looking at the options:

• A. -4: This is outside the range $[-3, \infty)$.
• B. -2: This is within the range.
• C. 0: This is within the range.
• D. 2: This is within the range.

Conclusion

The value that could not be in the range of $f(g(x))$ is: A. -4

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How would the solution change if $f(x)$ or $g(x)$ were modified?
2. What would be the range if $g(x)$ were a linear function instead?
3. How does the domain of $g(x)$ affect the possible values for $f(g(x))$?
4. Can you find a different function pair where a specific output is excluded from $f(g(x))$?
5. How does composition of functions affect the range in general?

Tip: When analyzing composite functions, start by focusing on the range of the inner function to help determine the output possibilities for the overall composition.