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!