Suppose 
f
1
(
x
)
f 
1
​
 (x) and 
f
2
(
x
)
f 
2
​
 (x) are functions defined on domains 
D
1
⊂
R
D 
1
​
 ⊂R and 
D
2
⊂
R
D 
2
​
 ⊂R, respectively and codomains are subset of 
R
R . What will be the domain of the function 
f
1
(
x
)
+
f
2
(
x
)
f 
1
​
 (x)+f 
2
​
 (x)?

D
1
∪
D
2
D 
1
​
 ∪D 
2
​
 

D
1
∖
D
2
D 
1
​
 ∖D 
2
​
 

D
2
∖
D
1
D 
2
​
 ∖D 
1
​
 

D
1
∩
D
2
D 
1
​
 ∩D 
2
​


The domain of the function \( f_1(x) + f_2(x) \) will be the set of all \( x \) values for which both \( f_1(x) \) and \( f_2(x) \) are defined. This means that the new function can only be evaluated where both functions are simultaneously defined. Therefore, the domain of the function \( f_1(x) + f_2(x) \) will be the **intersection** of the domains of \( f_1(x) \) and \( f_2(x) \), denoted by \( D_1 \cap D_2 \).

So, the correct answer is:

\[
D_1 \cap D_2
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the domain of \( f_1(x) \cdot f_2(x) \) determined?
2. What happens if \( f_1(x) \) or \( f_2(x) \) have undefined values in their domains?
3. How do we find the domain for the quotient \( \frac{f_1(x)}{f_2(x)} \)?
4. What is the domain of \( f_1(x) + f_2(x) \) if one of the functions is a constant?
5. How would the domain change if both functions were defined piecewise?

**Tip**: When combining functions, always consider where each individual function is defined, as the domain of the combination depends on both.

Suppose f1(x) and f2(x) are functions defined on domains D1 ⊂ R and D2 ⊂ R, respectively, and codomains are a subset of R. What will be the domain of the function f1(x) + f2(x)? D1 ∪ D2, D1 ∖ D2, D2 ∖ D1, or D1 ∩ D2?

Explore how to determine the domain of a function formed by adding two functions, f1(x) and f2(x). This problem highlights key concepts such as functions, domains, and set theory, with a focus on the intersection of domains for proper function evaluation. Ideal for students in Grades 10-12.

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