Find f​(g​(x​)) and g​(f​(x​)) and determine whether the pair of functions f and g are inverses of each other.
f left parenthesis x right parenthesis equals StartFraction 8 Over x minus 7 EndFraction
   and   g left parenthesis x right parenthesis equals StartFraction 8 Over x EndFraction plus 7

We are given the functions:

\[
f(x) = \frac{8}{x - 7}
\]
and
\[
g(x) = \frac{8}{x} + 7
\]

### Step 1: Find \( f(g(x)) \)
We substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f\left( \frac{8}{x} + 7 \right)
\]
Now, substitute \( \frac{8}{x} + 7 \) into the formula for \( f(x) \):

\[
f\left( \frac{8}{x} + 7 \right) = \frac{8}{\left( \frac{8}{x} + 7 \right) - 7}
\]

Simplify the denominator:

\[
f(g(x)) = \frac{8}{\frac{8}{x} + 7 - 7} = \frac{8}{\frac{8}{x}} = x
\]

So, \( f(g(x)) = x \).

### Step 2: Find \( g(f(x)) \)
We now substitute \( f(x) \) into \( g(x) \):

\[
g(f(x)) = g\left( \frac{8}{x - 7} \right)
\]
Substitute \( \frac{8}{x - 7} \) into the formula for \( g(x) \):

\[
g\left( \frac{8}{x - 7} \right) = \frac{8}{\frac{8}{x - 7}} + 7
\]

Simplify:

\[
g(f(x)) = x - 7 + 7 = x
\]

So, \( g(f(x)) = x \).

### Step 3: Conclusion
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f \) and \( g \) are **inverses** of each other.

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Here are 5 related questions for practice:
1. Find the inverse of \( f(x) = \frac{10}{x + 2} \).
2. Verify if \( f(x) = 2x + 1 \) and \( g(x) = \frac{x - 1}{2} \) are inverses.
3. Find \( f(g(x)) \) for \( f(x) = 3x - 4 \) and \( g(x) = \frac{x + 4}{3} \).
4. Prove that \( f(x) = \frac{1}{x} \) is its own inverse.
5. Solve \( f(g(x)) = 3 \) for \( f(x) = \frac{6}{x + 1} \) and \( g(x) = \frac{6}{x - 1} \).

**Tip:** If \( f(g(x)) = g(f(x)) = x \), then \( f \) and \( g \) are guaranteed to be inverses.

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) = 8/(x - 7) and g(x) = 8/x + 7

Learn how to verify if two functions, f(x) = 8/(x - 7) and g(x) = 8/x + 7, are inverses of each other by calculating f(g(x)) and g(f(x)). This solution demonstrates key algebraic concepts such as function composition and inverse functions, suitable for students in Grades 10-12.

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