https://file.aiquickdraw.com/mathbot/file/17242686149353wcfh0mu.jpg

Student number is 30 and g(x) is 0^-x the last digit is 0 

Thanks for the clarification. Let's work through the problem using the details provided, including your student number's last digit as 0 and \( g(x) = 0^{-x} \).

### 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 expression \( 0^{-x} \) is problematic because raising 0 to any negative power is undefined (it implies division by zero).
   - Thus, \( g(x) \) is undefined for all \( x \neq 0 \).
   - The domain for this function is undefined for all real numbers due to the presence of \( 0^{-x} \).

### b. \( f(x) \cdot p(x) \)
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 explanations on these results?

Here are five related questions to explore further:
1. What happens if you change the digit to something other than 0?
2. How would the function \( f(x) \) change if it involved a different radical?
3. Can you determine the range for each of these functions within their domains?
4. What would the graphs of these functions look like, and how do they behave near undefined points?
5. How does the undefined nature of \( 0^{-x} \) impact mathematical expressions and calculations?

**Tip:** Always analyze the domain restrictions carefully, especially when dealing with zero in exponents, logarithmic functions, and square roots to avoid undefined results.

Explore mathematical functions involving exponentials, trigonometrics, and logarithms. Includes detailed solutions and domain analyses for functions like g(x) = 0^{-x}, m(x) = \sin x, f(x) = \sqrt{x}, p(x) = \frac{1}{x-1}, and h(x) = \log(1-x). Suitable for advanced high school students and anyone interested in mathematical function analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation