To determine whether the given pair of functions are inverses, we need to check if:

• $f(g(x)) = x$
• $g(f(x)) = x$

Given:

• $g(x) = 4x + 1$
• $f(x) = -4x - 1$

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

Substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(4x + 1) = -4(4x + 1) - 1$

Simplify:

$f(g(x)) = -16x - 4 - 1 = -16x - 5$

Since $f(g(x)) \neq x$, the functions are not inverses based on this.

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

Substitute $f(x)$ into $g(x)$:

$g(f(x)) = g(-4x - 1) = 4(-4x - 1) + 1$

Simplify:

$g(f(x)) = -16x - 4 + 1 = -16x - 3$

Since $g(f(x)) \neq x$, the functions are also not inverses based on this check.

Conclusion:

The functions $g(x) = 4x + 1$ and $f(x) = -4x - 1$ are not inverses of each other.

Would you like more details on any step or further explanation?

Here are 5 related questions to expand your understanding:

1. How do you algebraically verify if two functions are inverses?
2. What is the geometric interpretation of inverse functions on a graph?
3. Can you find the inverse of $g(x) = 4x + 1$?
4. How can you check for inverse functions using compositions of functions?
5. What happens to the slopes of linear functions when they are inverses?

Tip: When finding the inverse of a function, swap $x$ and $y$ and then solve for $y$ to get the inverse.