We are given two functions, $f(x) = 2x + 1$ and $g(x) = 2x^2 - 1$, and asked to find $f(g(-2))$.

Here’s how we can solve this step by step:

1. First, find $g(-2)$: $g(x) = 2x^2 - 1$ Substituting $x = -2$: $g(-2) = 2(-2)^2 - 1 = 2(4) - 1 = 8 - 1 = 7$

2. Now that we know $g(-2) = 7$, we substitute $7$ into the function $f(x)$: $f(x) = 2x + 1$ So, $f(7) = 2(7) + 1 = 14 + 1 = 15$.

Thus, $f(g(-2)) = 15$.

Would you like more details or explanations?

Here are 5 follow-up questions you might explore:

1. What happens if $g(x)$ is a more complex polynomial?
2. Can you find $g(f(-2))$?
3. How would you approach solving this problem if the functions were given as trigonometric functions instead?
4. What would be the result if $f(x)$ were a logarithmic function and $g(x)$ stayed the same?
5. How does function composition affect the domain and range of a function?

Tip: When composing functions, always evaluate the innermost function first!