If the functions are combined so that h(x) = f(x) – g(x), then what is the domain of the function

To determine the domain of the function \( h(x) = f(x) - g(x) \), we need to consider the domains of both \( f(x) \) and \( g(x) \). The domain of \( h(x) \) will be the intersection of the domains of \( f(x) \) and \( g(x) \), as \( h(x) \) can only be defined where both \( f(x) \) and \( g(x) \) are defined.

### Steps:
1. **Determine the domain of \( f(x) \):** Identify all the values of \( x \) for which \( f(x) \) is defined.
2. **Determine the domain of \( g(x) \):** Identify all the values of \( x \) for which \( g(x) \) is defined.
3. **Find the intersection of these domains:** The domain of \( h(x) = f(x) - g(x) \) will be the set of \( x \) values that are common to both the domain of \( f(x) \) and the domain of \( g(x) \).

### Example:
- If \( f(x) \) is defined for all \( x \geq 0 \) (i.e., \( \text{Domain of } f(x) = [0, \infty) \))
- And \( g(x) \) is defined for all \( x \leq 5 \) (i.e., \( \text{Domain of } g(x) = (-\infty, 5] \))

The domain of \( h(x) = f(x) - g(x) \) would be \( [0, 5] \), which is the intersection of the two domains.

If \( f(x) \) and \( g(x) \) are both defined for all real numbers, then the domain of \( h(x) \) would also be all real numbers, \( (-\infty, \infty) \).

### Conclusion:
The domain of \( h(x) = f(x) - g(x) \) is the intersection of the domains of \( f(x) \) and \( g(x) \).

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### Relative Questions:
1. How do you find the domain of a function involving square roots?
2. What happens to the domain when you multiply two functions together?
3. Can the domain of a function be restricted by the range of another function it’s combined with?
4. How do you find the domain of a quotient of two functions \( f(x)/g(x) \)?
5. What is the domain of the sum of two functions \( f(x) + g(x) \)?

### Tip:
When combining functions, always check for restrictions in the domain, such as those caused by square roots, denominators, or logarithms.

Learn how to determine the domain of a function h(x) = f(x) - g(x) by finding the intersection of the domains of f(x) and g(x). This problem explores the concept of domain in functions and how to analyze and compute intersections of sets. Suitable for students in Grades 10-12.

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