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We are given two functions \( f \) and \( g \), and the problem asks to show that the composition \( f \circ g \) has the same domain as \( g \) if and only if the range of \( g \) is contained in the domain of \( f \).

### Definitions and Breakdown:
- **Domain of a function**: The set of all possible inputs for the function.
- **Range of a function**: The set of all possible outputs from the function.
- **Composition of functions**: For \( f \circ g \), the output of \( g(x) \) becomes the input for \( f(x) \). The composition \( f \circ g \) is only defined when the output of \( g(x) \) lies within the domain of \( f \).

### To Prove:

We need to prove that:
1. \( \text{Dom}(f \circ g) = \text{Dom}(g) \) if and only if the range of \( g \subseteq \text{Dom}(f) \).

#### Forward Direction (If \( \text{Dom}(f \circ g) = \text{Dom}(g) \), then Range of \( g \subseteq \text{Dom}(f) \)):

- Let \( x \in \text{Dom}(g) \), so \( g(x) \) is well-defined.
- For \( f \circ g \) to be well-defined for all \( x \in \text{Dom}(g) \), we must have \( f(g(x)) \) defined.
- This means that \( g(x) \) must lie in the domain of \( f \). Therefore, \( \text{Range}(g) \subseteq \text{Dom}(f) \).

#### Reverse Direction (If \( \text{Range}(g) \subseteq \text{Dom}(f) \), then \( \text{Dom}(f \circ g) = \text{Dom}(g) \)):

- Assume \( \text{Range}(g) \subseteq \text{Dom}(f) \), meaning for every \( x \in \text{Dom}(g) \), \( g(x) \in \text{Dom}(f) \).
- This ensures that \( f(g(x)) \) is well-defined for all \( x \in \text{Dom}(g) \).
- Thus, \( f \circ g \) is defined for all \( x \in \text{Dom}(g) \), implying \( \text{Dom}(f \circ g) = \text{Dom}(g) \).

Hence, the composition \( f \circ g \) has the same domain as \( g \) if and only if the range of \( g \) is contained in the domain of \( f \).

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### Five Related Questions:
1. What is the domain and range of a function, and how do they relate to function composition?
2. Can the composition of two functions ever result in a smaller domain than the original function?
3. How does the concept of injective and surjective functions relate to the composition of functions?
4. Can a function have a domain that is a proper subset of another function's domain?
5. How do we prove that a function is well-defined for a given input?

### Tip:
When working with function compositions, always ensure the output of the inner function lies within the domain of the outer function, otherwise the composition will be undefined.

Suppose f and g are functions. Show that the composition f∘g has the same domain as g if and only if the range of g is contained in the domain of f.

This problem demonstrates how to prove that the composition of functions f∘g has the same domain as g if and only if the range of g is contained in the domain of f. Detailed explanation with key concepts in function composition, domain, and range. Suitable for advanced high school or undergraduate students.

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