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Given the details you've provided, your student number is 30, so the last digit is 0. You've also mentioned that \( g(x) = 0^{-x} \). Let's proceed with solving the expressions and determining the domain for each.

### a. \( g(x) - m(x) \)
1. **Given functions:**
   - \( g(x) = 0^{-x} \)
   - \( m(x) = \sin x \) (since the last digit is 0, \( m(x) = \sin x \))

2. **Simplified form:**
   - \( g(x) - m(x) = 0^{-x} - \sin x \)

3. **Domain:**
   - The function \( g(x) = 0^{-x} \) is not valid because \( 0^{-x} \) (where \( x \neq 0 \)) leads to division by zero, which is undefined.
   - So, the domain is undefined.

### b. \( f(x)p(x) \)
1. **Given functions:**
   - \( f(x) = \sqrt{x} \)
   - \( p(x) = \frac{1}{x-1} \)

2. **Simplified form:**
   - \( f(x) \cdot p(x) = \sqrt{x} \cdot \frac{1}{x-1} = \frac{\sqrt{x}}{x-1} \)

3. **Domain:**
   - \( \sqrt{x} \) is defined for \( x \geq 0 \).
   - \( \frac{1}{x-1} \) is defined for \( x \neq 1 \).
   - Therefore, the domain is \( x \geq 0 \) and \( x \neq 1 \).

### c. \( \frac{h(x)}{p(x)} \)
1. **Given functions:**
   - \( h(x) = \log (1-x) \)
   - \( p(x) = \frac{1}{x-1} \)

2. **Simplified form:**
   - \( \frac{h(x)}{p(x)} = \frac{\log(1-x)}{\frac{1}{x-1}} = (x-1) \log(1-x) \)

3. **Domain:**
   - \( \log(1-x) \) is defined for \( 1-x > 0 \) which implies \( x < 1 \).
   - \( p(x) = \frac{1}{x-1} \) is defined for \( x \neq 1 \).
   - Therefore, the domain is \( x < 1 \).

Would you like any further details or have any questions?

Here are five related questions to explore further:
1. How does changing the last digit affect each expression and its domain?
2. What would happen if \( g(x) \) were different, for instance, \( g(x) = 2^{-x} \)?
3. How do logarithmic and sine functions compare in their domains and ranges?
4. How can the undefined nature of \( 0^{-x} \) be explained in different contexts?
5. How can we further simplify or analyze the expression \( (x-1) \log(1-x) \)?

**Tip:** Always check for restrictions in the domain, particularly when dealing with logarithmic, square root, or rational functions to avoid undefined expressions.

Explore solutions to mathematical expressions involving exponential functions, trigonometric functions, and rational functions. Learn about domain restrictions and the interplay between functions like sine, logarithmic, and square root functions. Suitable for advanced high school students and those preparing for college-level mathematics.

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