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The image contains functions \( f(x) = \frac{1}{x^2 + 10} \) and \( g(x) = \frac{1}{x^2 - 10} \), and it asks to find the following:

1. \( f + g \)
2. \( f - g \)
3. \( fg \)
4. \( \frac{f}{g} \)

Let's work through these step by step.

### 1. \( f + g \):
\[
f(x) + g(x) = \frac{1}{x^2 + 10} + \frac{1}{x^2 - 10}
\]
To add these fractions, we need a common denominator:
\[
f(x) + g(x) = \frac{(x^2 - 10) + (x^2 + 10)}{(x^2 + 10)(x^2 - 10)} = \frac{2x^2}{(x^4 - 100)}
\]
Thus,
\[
f(x) + g(x) = \frac{2x^2}{x^4 - 100}
\]

### 2. \( f - g \):
\[
f(x) - g(x) = \frac{1}{x^2 + 10} - \frac{1}{x^2 - 10}
\]
Using the common denominator as before:
\[
f(x) - g(x) = \frac{(x^2 - 10) - (x^2 + 10)}{(x^2 + 10)(x^2 - 10)} = \frac{-20}{x^4 - 100}
\]
Thus,
\[
f(x) - g(x) = \frac{-20}{x^4 - 100}
\]

### 3. \( fg \):
\[
fg = \frac{1}{x^2 + 10} \times \frac{1}{x^2 - 10} = \frac{1}{(x^2 + 10)(x^2 - 10)}
\]
Since \( (x^2 + 10)(x^2 - 10) = x^4 - 100 \), we have:
\[
fg = \frac{1}{x^4 - 100}
\]

### 4. \( \frac{f}{g} \):
\[
\frac{f(x)}{g(x)} = \frac{\frac{1}{x^2 + 10}}{\frac{1}{x^2 - 10}} = \frac{x^2 - 10}{x^2 + 10}
\]
Thus,
\[
\frac{f}{g} = \frac{x^2 - 10}{x^2 + 10}
\]

Would you like further details on any of the steps?

Here are five related questions to expand your understanding:
1. How would the results change if \( f(x) \) and \( g(x) \) had different powers of \( x \)?
2. Can you factor \( x^4 - 100 \) into simpler components?
3. What happens if \( x^2 = 10 \)? Are there any restrictions on \( x \)?
4. How do these operations affect the domains of \( f \) and \( g \)?
5. Can you graph these combined functions and interpret their intersections?

**Tip:** Always check for any restrictions in the domain when performing operations with functions. For instance, ensure that denominators are not zero!

Find the functions f + g, f - g, fg, and f/g. Given f(x) = 1 / (x^2 + 10) and g(x) = 1 / (x^2 - 10).

Learn how to perform operations on rational functions with the given f(x) = 1 / (x^2 + 10) and g(x) = 1 / (x^2 - 10). This problem covers key algebraic concepts like function addition, subtraction, multiplication, and division, suitable for students in Grades 9-12.

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