We are given two functions:

• $f(x) = 3|x + 1| - 4$
• $g(x) = -2x^2 - 5$

We are asked to find $f(g(x))$, which means we will substitute $g(x)$ into the function $f(x)$.

Step-by-step solution:

1. $g(x) = -2x^2 - 5$, so we substitute $g(x)$ into $f(x)$: $f(g(x)) = 3 \left| g(x) + 1 \right| - 4$ Substituting $g(x) = -2x^2 - 5$: $f(g(x)) = 3 \left| (-2x^2 - 5) + 1 \right| - 4$

2. Simplify the expression inside the absolute value: $(-2x^2 - 5) + 1 = -2x^2 - 4$ So, we have: $f(g(x)) = 3 \left| -2x^2 - 4 \right| - 4$

3. Since the absolute value of $-2x^2 - 4$ is $2x^2 + 4$ (because the absolute value removes the negative sign), we have: $f(g(x)) = 3(2x^2 + 4) - 4$

4. Now, distribute the 3: $f(g(x)) = 6x^2 + 12 - 4$

5. Simplify the expression: $f(g(x)) = 6x^2 + 8$

Thus, $f(g(x)) = 6x^2 + 8$.

Would you like any further clarifications?

Here are five related questions to further your understanding:

1. What happens if you change the sign inside the absolute value?
2. How would the answer change if $g(x)$ were linear instead of quadratic?
3. Can you apply this process to a different pair of functions, say $f(x) = x^2$ and $g(x) = x + 1$?
4. What is the significance of the absolute value in the function $f(x)$?
5. How would you differentiate $f(g(x))$ with respect to $x$?

Tip:

Always handle absolute value expressions carefully by considering both positive and negative cases of the content inside the absolute value.