given f(x) = 4/x and g(x) x+5/x-2 express the domain of (g*f)(x)in interval notation

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.

Given f(x) = 4/x and g(x) = (x+5)/(x-2), express the domain of (g*f)(x) in interval notation.

Learn how to find the domain of the composite function (g*f)(x) where f(x) = 4/x and g(x) = (x+5)/(x-2). This problem explores key concepts in rational functions and composite function domains. Suitable for students in Grades 10-12.

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