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We are given two functions:

1. \( f(x) = \frac{2 - x}{\sqrt{2x - 4}} \)
2. \( g(x) = \frac{x - 3}{x - 4} \)

We need to determine the domain of the function \( h(x) = f(x) + g(x) \).

### Step 1: Find the domain of \( f(x) \)

The domain of \( f(x) \) requires that:
- The expression inside the square root, \( 2x - 4 \), must be greater than 0 because the square root function is defined for positive numbers.
\[
2x - 4 > 0 \implies 2x > 4 \implies x > 2
\]
- Additionally, the denominator in \( f(x) \) is \( \sqrt{2x - 4} \), so \( 2x - 4 \neq 0 \). This confirms that \( x = 2 \) is not included in the domain.

Thus, the domain of \( f(x) \) is \( x > 2 \).

### Step 2: Find the domain of \( g(x) \)

The domain of \( g(x) \) requires that the denominator \( x - 4 \) is not zero:
\[
x - 4 \neq 0 \implies x \neq 4
\]
Thus, \( x \neq 4 \).

### Step 3: Combine the domain restrictions for \( h(x) = f(x) + g(x) \)

The domain of \( h(x) \) will be the intersection of the domains of \( f(x) \) and \( g(x) \). The domain of \( f(x) \) is \( x > 2 \), and the domain of \( g(x) \) is all real numbers except \( x = 4 \).

Therefore, the domain of \( h(x) \) is:
\[
x > 2 \quad \text{and} \quad x \neq 4
\]

### Final Answer:

The domain of \( h(x) = f(x) + g(x) \) is \( (2, 4) \cup (4, \infty) \).

Would you like further details or have any questions?

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Here are 5 related questions:
1. What is the range of the function \( h(x) = f(x) + g(x) \)?
2. How do we handle square roots when solving for domains?
3. What is the behavior of \( h(x) \) near \( x = 4 \)?
4. Can the function \( f(x) \) or \( g(x) \) have any discontinuities?
5. How would the domain change if the functions were composed as \( f(g(x)) \)?

**Tip**: Always consider both the numerator and denominator when determining the domain of a rational function.

Jika f(x) = (2 - x)/√(2x - 4) dan g(x) = (x - 3)/(x - 4), domain dari fungsi h(x) = f(x) + g(x) adalah ....

Learn how to find the domain of h(x) = f(x) + g(x) where f(x) = (2 - x)/√(2x - 4) and g(x) = (x - 3)/(x - 4). This problem covers algebraic concepts such as rational functions and the domain of square root functions. Suitable for students in Grades 10-12.

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