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 $f$.

Would you like more detailed examples or further explanation?

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.