Let's verify algebraically if $f(x) = \frac{x - 2}{7}$ and $g(x) = 7x + 2$ are inverses of each other.

To do this, we need to check the following two conditions:

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

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

Substitute $g(x) = 7x + 2$ into $f(x) = \frac{x - 2}{7}$:

$f(g(x)) = f(7x + 2) = \frac{(7x + 2) - 2}{7}$

Simplify the numerator:

$f(g(x)) = \frac{7x}{7} = x$

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

Substitute $f(x) = \frac{x - 2}{7}$ into $g(x) = 7x + 2$:

$g(f(x)) = g\left(\frac{x - 2}{7}\right) = 7\left(\frac{x - 2}{7}\right) + 2$

Simplify the multiplication:

$g(f(x)) = (x - 2) + 2 = x$

Conclusion:

Both conditions are satisfied: $f(g(x)) = x \quad \text{and} \quad g(f(x)) = x$

Thus, $f(x)$ and $g(x)$ are inverse functions of each other.

Would you like me to explain any steps in more detail?

Related Questions:

1. What are the steps for finding the inverse of a function algebraically?
2. How can you graphically verify that two functions are inverses?
3. What are the properties of inverse functions?
4. How does composition of functions relate to inverses?
5. Can you have a function that is its own inverse?

Tip:

When verifying inverse functions, always simplify each composition fully to check that it equals $x$.