We are given two functions:

• $f(x) = \frac{4}{x}$
• $g(x) = \frac{x+5}{x-2}$

We are asked to find the domain of the composite function $(g \circ f)(x)$, which means $(g \circ f)(x) = g(f(x))$. The domain of $(g \circ f)(x)$ will be determined by the restrictions on both functions.

Step 1: Find the domain of $f(x)$

The function $f(x) = \frac{4}{x}$ is undefined when $x = 0$ because division by zero is undefined.

Thus, the domain of $f(x)$ is:

$\text{Domain of } f(x) = (-\infty, 0) \cup (0, \infty)$

Step 2: Find the domain of $g(f(x))$

Now, we substitute $f(x)$ into $g(x)$:

$g(f(x)) = g\left(\frac{4}{x}\right) = \frac{\frac{4}{x} + 5}{\frac{4}{x} - 2}$

For $g(f(x))$ to be defined, the denominator must not be zero. So, we need to solve:

$\frac{4}{x} - 2 = 0$

Solving for $x$:

$\frac{4}{x} = 2 \implies x = 2$

Thus, $x = 2$ makes the denominator zero, and $g(f(x))$ is undefined at $x = 2$.

Step 3: Combine restrictions

We already know that $f(x)$ is undefined at $x = 0$ and now we also know that $g(f(x))$ is undefined at $x = 2$.

Thus, the domain of $(g \circ f)(x)$ is all real numbers except $x = 0$ and $x = 2$.

Final Answer:

The domain of $(g \circ f)(x)$ in interval notation is:

$(-\infty, 0) \cup (0, 2) \cup (2, \infty)$

Would you like more details on any of the steps, or have any questions?

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Here are 5 related questions:

1. What is the domain of $g(x)$ by itself?
2. How do you find the range of a composite function like $(g \circ f)(x)$?
3. What would happen if $f(x)$ had more restrictions in its domain?
4. How can you determine the domain of the sum of two functions, say $f(x) + g(x)$?
5. What is the domain of the product of two functions, $f(x) \cdot g(x)$?

Tip: Always check both the inner and outer functions when finding the domain of a composite function.