Let f(x) = √x and g(x) = x^2. Which statement about the domains of f∘g and g∘f is true?

Let f(x) = √x and g(x) = x^2. Which statement about the domains of f∘g and g∘f is true?

• A. f∘g is defined for all real numbers; g∘f is defined only for nonnegative x.
• B. f∘g is defined for all real numbers; g∘f is defined for all real numbers.
• C. Both compositions are defined only for x ≥ 0.
• D. Neither composition is defined for negative x.

Answer: (A)

When forming a composition, you must have the inner function defined at x and its output must lie in the domain of the outer function. For f∘g, the inner function is g(x) = x^2, which is defined for all real x and outputs x^2 ≥ 0. The outer function f has domain x ≥ 0, so since x^2 is always nonnegative, the inner output always fits the outer domain. Hence f∘g is defined for every real x. For g∘f, the inner function is f(x) = √x, which requires x ≥ 0. The outer function g accepts any real input, but here its input is √x, which is defined only when x ≥ 0. Therefore the domain of g∘f is x ≥ 0.

When forming a composition, you must have the inner function defined at x and its output must lie in the domain of the outer function.

For f∘g, the inner function is g(x) = x^2, which is defined for all real x and outputs x^2 ≥ 0. The outer function f has domain x ≥ 0, so since x^2 is always nonnegative, the inner output always fits the outer domain. Hence f∘g is defined for every real x.

For g∘f, the inner function is f(x) = √x, which requires x ≥ 0. The outer function g accepts any real input, but here its input is √x, which is defined only when x ≥ 0. Therefore the domain of g∘f is x ≥ 0.