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The problem asks to find \((f \circ g)(x)\), which represents the composition of functions \(f(x)\) and \(g(x)\). This means we need to substitute \(g(x)\) into \(f(x)\).

Given:
- \(f(x) = \frac{x}{x + 5}\)
- \(g(x) = 8x - 5\)

To find \((f \circ g)(x)\), we substitute \(g(x)\) into \(f(x)\):

\[
(f \circ g)(x) = f(g(x)) = f(8x - 5)
\]

Now substitute \(8x - 5\) into \(f(x)\):

\[
f(8x - 5) = \frac{8x - 5}{(8x - 5) + 5}
\]

Simplify the denominator:

\[
f(8x - 5) = \frac{8x - 5}{8x}
\]

Thus, the composition of the functions is:

\[
(f \circ g)(x) = \frac{8x - 5}{8x}
\]

Let me know if you would like further details or clarifications! Here are five related questions to explore:

1. What is the domain of the composed function \((f \circ g)(x)\)?
2. How would the result change if \(g(x)\) was a different linear function?
3. What is the inverse of the composed function \((f \circ g)(x)\)?
4. How do you find \((g \circ f)(x)\) for the same \(f(x)\) and \(g(x)\)?
5. Can you compute the value of \((f \circ g)(2)\)?

**Tip:** Always check the domain restrictions when composing functions, as certain values of \(x\) might lead to undefined expressions (e.g., division by zero).

Find (f o g)(x) for f(x) = x/(x + 5) and g(x) = 8x - 5.

Learn how to solve a composition of functions problem by finding (f o g)(x) for f(x) = x/(x + 5) and g(x) = 8x - 5. Detailed step-by-step solution with function substitution and algebraic simplifications. Suitable for Grades 10-12.

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