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.