To determine if $f(x)$ is the inverse of $g(x)$, we need to check if both functions undo each other. Specifically, we need to see if:

1. $f(g(x)) = x$
2. $g(f(x)) = x$

Let's break this down.

Given functions:

• $f(x) = -4(4x - 7)^2$, with $x \geq \frac{7}{4}$
• $g(x) = \sqrt{x} + 3$

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

Substitute $g(x)$ into $f(x)$: $f(g(x)) = f(\sqrt{x} + 3) = -4(4(\sqrt{x} + 3) - 7)^2$ Simplify inside the parentheses: $= -4(4\sqrt{x} + 12 - 7)^2 = -4(4\sqrt{x} + 5)^2$

This expression is not simplified to just $x$, which means $f(g(x)) \neq x$.

Step 2: Find $g(f(x))$

Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(-4(4x - 7)^2) = \sqrt{-4(4x - 7)^2} + 3$ Since the square root of a negative number is not defined for real numbers, this means $g(f(x))$ is not a valid expression for real $x$.

Conclusion:

Because neither $f(g(x)) = x$ nor $g(f(x)) = x$, we conclude that $f(x)$ is not the inverse of $g(x)$.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How do you find the inverse of a function algebraically?
2. What is the domain and range of a square root function like $g(x)$?
3. How can you graphically verify if two functions are inverses?
4. Why is it important to check both compositions when determining inverses?
5. What are some common restrictions on the domain for inverse functions?

Tip: When determining if two functions are inverses, always check both compositions, $f(g(x))$ and $g(f(x))$. They must both simplify to $x$.