Questions: Which description best explains the domain of (g ∘ 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

Solution Steps

The composition of functions \((g \circ f)(x)\) means that you first apply the function \(f\) to \(x\), and then apply the function \(g\) to the result of \(f(x)\). Therefore, for \((g \circ f)(x)\) to be defined, \(f(x)\) must be in the domain of \(g\).

The domain of \((g \circ f)(x)\) consists of all \(x\) values that are in the domain of \(f\) such that \(f(x)\) is in the domain of \(g\). This means that we need to consider the elements in the domain of \(f(x)\) for which \(g(f(x))\) is defined.

• The first option states: "the elements in the domain of \(f(x)\) for which \(g(f(x))\) is defined." This matches our understanding from Step 2.
• The second option states: "the elements in the domain of \(f(x)\) for which \(g(f(x))\) is not zero." This is incorrect because the domain is not concerned with whether \(g(f(x))\) is zero, but whether it is defined.
• The third option states: "the elements in the domain of \(g(x)\) for which \(g(f(x))\) is defined." This is incorrect because it does not consider the domain of \(f(x)\).
• The fourth option states: "the elements in the domain of \(g(x)\) for which \(g(f(x))\) is not zero." This is incorrect for the same reason as the second option.

Final Answer