for the functions f(x)=1/x-4 and g(x)= 13/x, find the composition f\:\circ \:g and simplify your answer as much as possible. Write the domain using interval notation.

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.

For the functions f(x) = 1/x - 4 and g(x) = 13/x, find the composition f ∘ g and simplify your answer as much as possible. Write the domain using interval notation.

Learn how to find the composition of f(x) = 1/x - 4 and g(x) = 13/x and simplify the result. The solution includes domain analysis using interval notation and step-by-step simplification of the function composition. Suitable for Grades 10-12.

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