Given two functions f and g with domains A and B. The domain of f ◦ g is the set of all real numbers x such that (i) x is in A and f(x) in B (ii) x is in both A and B (iii) x is in A or x is in B (iv) x is in B and g(x) in A (v) x is in A and g(x) in B (vi) none of the above

The domain of the composition of functions \( f \circ g \) (i.e., \( (f \circ g)(x) = f(g(x)) \)) is the set of all real numbers \( x \) such that:

1. \( x \) is in the domain of \( g \) (which means \( x \in B \)),
2. \( g(x) \) is in the domain of \( f \) (which means \( g(x) \in A \)).

This ensures that \( g(x) \) is defined, and \( f(g(x)) \) is also defined.

Thus, the correct choice is:

**(iv) \( x \) is in \( B \) and \( g(x) \) is in \( A \).**

Would you like further details or have any other questions? Here are some related questions you might find interesting:

1. How do you find the range of the composition of two functions?
2. What happens to the domain of \( f \circ g \) if \( g(x) \) is not defined for certain values of \( x \)?
3. Can the domain of \( f \circ g \) ever be larger than the domain of \( g \)?
4. What is the effect on the domain of \( f \circ g \) if \( f(x) \) has restrictions like square roots or logarithms?
5. How does the domain of the composition compare to the intersection of the domains of \( f \) and \( g \)?

**Tip:** When working with compositions, always start by identifying the domain of the inner function and then determine where the outer function is defined based on the output of the inner function.

Explore the domain of composition of functions with detailed explanations and examples. Learn how to determine the domain of \( f \circ g \) where \( f \) and \( g \) are functions with specified domains. Suitable for high school students and anyone studying functions and their compositions.

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