https://file.aiquickdraw.com/mathbot/file/1724267013711i0cdgi9v.jpg

Thanks for sharing the image again. I see that you have updated information that \( g(x) = 0^{-x} \) for question 2. Let's analyze the problems with this updated function.

### Question 2 Analysis

Given:
- \( f(x) = x^x \)
- \( g(x) = 0^{-x} \)
- \( h(x) = \log(x - 1) \)
- \( m(x) = x^3 \)

Let's address each part of the question.

#### **Part a: Simplify \( g(x) - m(x) \) and find the domain.**

- \( g(x) = 0^{-x} \)
- \( m(x) = x^3 \)

The expression becomes:
\[
g(x) - m(x) = 0^{-x} - x^3
\]

##### Simplification:
\( 0^{-x} \) is undefined for any real number \( x \), because raising 0 to any negative power results in division by zero, which is undefined. Therefore, the function \( g(x) = 0^{-x} \) itself is undefined for all real \( x \).

**Conclusion:** \( g(x) - m(x) \) is undefined for all real \( x \), so the domain is empty.

#### **Part b: Simplify \( f(h(x)) \) and find the domain.**

- \( f(x) = x^x \)
- \( h(x) = \log(x - 1) \)

The expression becomes:
\[
f(h(x)) = (\log(x - 1))^{\log(x - 1)}
\]

##### Domain:
- \( \log(x - 1) \) is defined when \( x - 1 > 0 \) or \( x > 1 \).
- \( f(x) = x^x \) is typically defined when \( x > 0 \), since for \( x \leq 0 \), the expression \( x^x \) can be undefined or complex.

Thus, the domain for \( f(h(x)) \) is \( x > 1 \).

#### **Part c: Simplify \( h(g(x)) \) and find the domain.**

- \( h(x) = \log(x - 1) \)
- \( g(x) = 0^{-x} \)

The expression becomes:
\[
h(g(x)) = \log(0^{-x} - 1)
\]

##### Simplification:
Since \( g(x) = 0^{-x} \) is undefined for all real \( x \), \( h(g(x)) \) is also undefined for all real \( x \). Thus, the domain is empty.

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Would you like to proceed with more details or have any questions? Let me know if you need further clarification or assistance with another part.

### Related Questions:
1. What does it mean for a function to be "undefined" at certain values of \( x \)?
2. How does the concept of a function's domain affect its graph?
3. What are some examples of functions that are undefined for all real numbers?
4. How can logarithmic functions affect the domain of composite functions?
5. How do you handle operations with zero in the context of exponential functions?

**Tip**: Always carefully evaluate the domain of each component function when working with compositions or combinations of functions.

Explore the detailed analysis of functions f(x), g(x), h(x), and m(x) with their compositions and domains. Understand why functions g(x) and h(g(x)) are undefined for all real numbers. Suitable for advanced high school students studying advanced functions and their domains.

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