Question
Which description best explains the domain of \( (g \circ f)(x) \) ? the elements in the domain of \( f(x) \) for which \( g(f(x)) \) is defined the elements in the domain of \( f(x) \) for which \( g(f(x)) \) is not zero the elements in the domain of \( g(x) \) for which \( g(f(x)) \) is defined the elements in the domain of \( g(x) \) for which \( g(f(x)) \) is not zero
Ask by Lee Nunez. in the United States
Mar 21,2025
Upstudy AI Solution
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Answer
The domain of \( (g \circ f)(x) \) includes all \( x \) in the domain of \( f(x) \) where \( g(f(x)) \) is defined.
Solution
1. The domain of the composite function \( (g \circ f)(x) \) consists of all values \( x \) such that:
- \( x \) is in the domain of \( f(x) \), and
- \( f(x) \) is in the domain of \( g(x) \).
2. This means that for every \( x \) we choose, \( g(f(x)) \) must be defined.
3. Therefore, the best description is:
- The elements in the domain of \( f(x) \) for which \( g(f(x)) \) is defined.
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Simplify this solution The Deep Dive
The domain of \( (g \circ f)(x) \) consists of the elements in the domain of \( f(x) \) for which \( g(f(x)) \) is defined. This means any input \( x \) you choose must first be valid for \( f \), and the output \( f(x) \) must also fall within the domain of \( g \) to ensure \( g(f(x)) \) can be computed. To put it simply, when chaining two functions, it's crucial that the output of the first function (in this case, \( f \)) must be a valid input for the second function (i.e., \( g \)). So think of it as a relay race—each runner (function) must stay in their lane and pass the baton (output) smoothly!
