To find the composition $f \circ g$, we evaluate $f(g(x))$, which means we substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = \frac{1}{x} - 4$
• $g(x) = \frac{13}{x}$

The composition $f \circ g$ is:

$f(g(x)) = f\left(\frac{13}{x}\right)$

Substitute $g(x) = \frac{13}{x}$ into $f(x) = \frac{1}{x} - 4$:

$f\left(\frac{13}{x}\right) = \frac{1}{\frac{13}{x}} - 4$

Simplify the fraction:

$\frac{1}{\frac{13}{x}} = \frac{x}{13}$

Thus:

$f(g(x)) = \frac{x}{13} - 4$

So, the simplified form of $f \circ g(x)$ is:

$f \circ g(x) = \frac{x}{13} - 4$

Domain:

The domain of $f \circ g(x)$ is determined by the domain of $g(x)$ and any restrictions introduced by $f(x)$.

1. Domain of $g(x)$: $g(x) = \frac{13}{x}$ is undefined when $x = 0$.
2. Domain of $f(x)$: $f(x) = \frac{1}{x} - 4$ is undefined when $x = 0$.

Since $g(x) = \frac{13}{x}$ is undefined at $x = 0$, $f(g(x))$ is also undefined at $x = 0$.

Therefore, the domain of $f \circ g(x)$ in interval notation is:

$(-\infty, 0) \cup (0, \infty)$

Final Answer:

• Composition: $f \circ g(x) = \frac{x}{13} - 4$
• Domain: $(-\infty, 0) \cup (0, \infty)$

Would you like more details, or do you have any questions?

Here are some related questions you might find interesting:

1. How would the composition change if $f(x) = \frac{2}{x} - 5$ and $g(x) = \frac{7}{x}$?
2. What would the domain be if $f(x) = \frac{1}{x-1}$ and $g(x) = \frac{13}{x}$?
3. How does the domain of $g(x)$ affect the composition $f \circ g(x)$?
4. Can you find the inverse function of $f \circ g(x)$?
5. What happens to $f \circ g(x)$ if you shift $g(x)$ horizontally?

Tip: When dealing with function compositions, always consider the domains of the inner and outer functions to avoid undefined values in the final expression.