To verify if the given pairs of functions are inverses of each other, we need to check if the composition of the two functions, $f(g(x))$ and $g(f(x))$, equals $x$ for each pair.

Let's go through each pair step by step.

1. Pair 1:

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

Step 1: Compute $f(g(x))$. $f(g(x)) = 3\left(\frac{1}{3}x - 2\right) + 6 = x - 6 + 6 = x$

Step 2: Compute $g(f(x))$. $g(f(x)) = \frac{1}{3}(3x + 6) - 2 = x + 2 - 2 = x$

Since $f(g(x)) = x$ and $g(f(x)) = x$, the functions $f(x)$ and $g(x)$ are inverses of each other.

2. Pair 2:

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

Step 1: Compute $f(g(x))$. $f(g(x)) = -2\left(\frac{-1}{2}x + 3\right) - 6 = x - 6 - 6 = x - 12$ $f(g(x)) \neq x$.

Step 2: Compute $g(f(x))$. $g(f(x)) = \frac{-1}{2}(-2x - 6) + 3 = x + 3 + 3 = x + 6$ $g(f(x)) \neq x$.

Since neither $f(g(x)) = x$ nor $g(f(x)) = x$, the functions $f(x)$ and $g(x)$ are not inverses of each other.

3. Pair 3:

$f(x) = -2x + 4$ $g(x) = \frac{-1}{2}x + 2$

Step 1: Compute $f(g(x))$. $f(g(x)) = -2\left(\frac{-1}{2}x + 2\right) + 4 = x - 4 + 4 = x$

Step 2: Compute $g(f(x))$. $g(f(x)) = \frac{-1}{2}(-2x + 4) + 2 = x - 2 + 2 = x$

Since $f(g(x)) = x$ and $g(f(x)) = x$, the functions $f(x)$ and $g(x)$ are inverses of each other.

4. Pair 4:

$f(x) = -\frac{3}{5}x - 10$ $g(x) = \frac{-5}{3}x + 12$

Step 1: Compute $f(g(x))$. $f(g(x)) = -\frac{3}{5}\left(\frac{-5}{3}x + 12\right) - 10 = x - 36 - 10 = x - 46$ $f(g(x)) \neq x$.

Step 2: Compute $g(f(x))$. $g(f(x)) = \frac{-5}{3}\left(-\frac{3}{5}x - 10\right) + 12 = x + 50 + 12 = x + 62$ $g(f(x)) \neq x$.

Since neither $f(g(x)) = x$ nor $g(f(x)) = x$, the functions $f(x)$ and $g(x)$ are not inverses of each other.

Summary:

• Pair 1 and Pair 3 are inverses of each other.
• Pair 2 and Pair 4 are not inverses of each other.

Would you like any more details on the steps or have any other questions?

Here are five related questions you might find helpful:

1. How can you algebraically find the inverse of a function?
2. What are the conditions for two functions to be inverses of each other?
3. Can a function have more than one inverse? Why or why not?
4. How can you graphically verify if two functions are inverses?
5. What happens if you compose a function with its inverse?

Tip: When verifying if two functions are inverses, always ensure that both compositions $f(g(x))$ and $g(f(x))$ simplify to $x$. If either one does not, the functions are not inverses.