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We are given two functions \( f(x) \) and \( g(x) \) and asked to compute \( (f \circ g)(x) \), which represents the composition of the functions \( f \) and \( g \).

The functions provided are:
\[
f(x) = \frac{1}{-5x + 10}
\]
\[
g(x) = \frac{1}{x^2 + 2x + 5}
\]

The composition \( (f \circ g)(x) \) is calculated as:
\[
(f \circ g)(x) = f(g(x))
\]
This means we substitute \( g(x) \) into \( f(x) \).

1. Start with \( f(x) = \frac{1}{-5x + 10} \).
2. Replace \( x \) in \( f(x) \) with \( g(x) = \frac{1}{x^2 + 2x + 5} \).

So,
\[
(f \circ g)(x) = \frac{1}{-5 \cdot \left( \frac{1}{x^2 + 2x + 5} \right) + 10}
\]

Now simplify the expression:
\[
(f \circ g)(x) = \frac{1}{-\frac{5}{x^2 + 2x + 5} + 10}
\]

To simplify further, get a common denominator in the denominator:
\[
(f \circ g)(x) = \frac{1}{\frac{-5 + 10(x^2 + 2x + 5)}{x^2 + 2x + 5}}
\]
\[
(f \circ g)(x) = \frac{1}{\frac{-5 + 10x^2 + 20x + 50}{x^2 + 2x + 5}}
\]
\[
(f \circ g)(x) = \frac{1}{\frac{10x^2 + 20x + 45}{x^2 + 2x + 5}}
\]

Finally, invert the denominator to complete the simplification:
\[
(f \circ g)(x) = \frac{x^2 + 2x + 5}{10x^2 + 20x + 45}
\]

Thus, the composition \( (f \circ g)(x) \) is:
\[
(f \circ g)(x) = \frac{x^2 + 2x + 5}{10x^2 + 20x + 45}
\]

Would you like a more detailed explanation of any step? 

Here are 5 related questions to further explore:

1. What is the domain of the composite function \( (f \circ g)(x) \)?
2. How do you compute the inverse of a composite function like \( (f \circ g)(x) \)?
3. What happens if you switch the composition, i.e., calculate \( (g \circ f)(x) \)?
4. How do you simplify complex rational functions like \( f(g(x)) \)?
5. What are the applications of function composition in real-world problems?

**Tip:** When performing function compositions, always replace the inner function first and simplify step by step.

Suppose f(x) = 1/(-5x + 10) and g(x) = 1/(x^2 + 2x + 5); Find (f o g)(x).

Learn how to compute the composition of two rational functions, f(x) and g(x), where f(x) = 1/(-5x + 10) and g(x) = 1/(x^2 + 2x + 5). This detailed solution covers function composition, algebraic manipulation, and simplification. Suitable for students in Grades 10-12.

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