Let f and g be functions.
(a)
The function 
(f + g)(x)
 is defined for all values of x that are in the domain(s) of?

The function \((f + g)(x)\) is defined for all values of \(x\) that are in the domain of both \(f(x)\) and \(g(x)\). In other words, \((f + g)(x)\) is defined on the **intersection of the domains** of \(f(x)\) and \(g(x)\). This means:

- For every value \(x\), both \(f(x)\) and \(g(x)\) must be defined in order for \((f + g)(x)\) to be defined.

### To summarize:
- \((f + g)(x)\) is defined for all \(x\) such that \(x \in \text{Domain}(f) \cap \text{Domain}(g)\).

Would you like further details or any clarifications?

Here are 5 related questions to explore this further:
1. What happens if the domains of \(f(x)\) and \(g(x)\) do not overlap at all?
2. How do you find the domain of the product \((f \cdot g)(x)\)?
3. What is the domain of the composition of functions \((f \circ g)(x)\)?
4. How does the domain change if either \(f(x)\) or \(g(x)\) is undefined at certain points?
5. Can the range of \((f + g)(x)\) be greater than the range of both \(f(x)\) and \(g(x)\)?

**Tip**: When adding two functions, always check the domains of both to ensure they overlap at the values you're interested in!

Let f and g be functions. (a) The function (f + g)(x) is defined for all values of x that are in the domain(s) of?

Explore how to determine the domain of the sum of functions (f + g)(x). This explanation covers the intersection of the domains of f(x) and g(x), providing a clear understanding for high school students. Suitable for Grades 9-12.

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