We are given two functions $f(x) = 2x - 7$ and $g(x) = \frac{x + 2}{7}$. To determine if $f$ and $g$ are inverses of each other, we need to:

1. Find $f(g(x))$.
2. Find $g(f(x))$.
3. Check if $f(g(x)) = x$ and $g(f(x)) = x$. If both are true, then $f$ and $g$ are inverses.

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

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

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

Simplifying:

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

Now, distribute the 2:

$f(g(x)) = \frac{2x + 4}{7} - 7$

Convert 7 to a fraction with denominator 7:

$f(g(x)) = \frac{2x + 4}{7} - \frac{49}{7}$

Combine the terms:

$f(g(x)) = \frac{2x + 4 - 49}{7} = \frac{2x - 45}{7}$

Thus, $f(g(x)) = \frac{2x - 45}{7}$, which is not equal to $x$.

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

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

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

Simplifying:

$g(f(x)) = \frac{2x - 7 + 2}{7} = \frac{2x - 5}{7}$

Thus, $g(f(x)) = \frac{2x - 5}{7}$, which is not equal to $x$.

Conclusion:

Since neither $f(g(x)) = x$ nor $g(f(x)) = x$, the functions $f(x) = 2x - 7$ and $g(x) = \frac{x + 2}{7}$ are not inverses of each other.

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Do you need more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How can you tell if two functions are inverses using their compositions?
2. What would the function inverse of $f(x) = 2x - 7$ look like?
3. Why is the condition $f(g(x)) = x$ important for determining inverses?
4. Can two functions be partially inverse (e.g., for some but not all values)?
5. How do graphical methods help in identifying inverse functions?

Tip: When composing functions, carefully apply each function's rule and simplify step by step to avoid errors in the process.