Find f​(g​(x​)) 
and g​(f​(x​)) 
and determine whether the pair of functions f and g are inverses of each other.
f​(x​)equals 2 x minus 7
   
and   g​(x​)equals StartFraction x plus 2 Over 7 EndFraction

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**.

---

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.

Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other. f(x) equals 2x - 7 and g(x) equals (x + 2) / 7.

Learn how to find f(g(x)) and g(f(x)) to determine if two functions are inverses. This example uses f(x) = 2x - 7 and g(x) = (x + 2) / 7. Step-by-step solution explains key algebraic concepts such as function composition and the inverse function theorem. Suitable for students in Grades 9-11.

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