Determine f∘g(x) and its domain given f(x)=√(x+4) and g(x)=x^2.

Determine f∘g(x) and its domain given f(x)=√(x+4) and g(x)=x^2.

• A. f∘g(x) = √(x^2+4); domain all real numbers.
• B. f∘g(x) = √(x+4) + x^2
• C. f∘g(x) = √(x^2+4x)
• D. f∘g(x) = √(x^2 - 4)

Answer: (A)

Composing means you feed g(x) into f. With f(u) = √(u + 4) and g(x) = x^2, you get f∘g(x) = √(g(x) + 4) = √(x^2 + 4). The domain is determined by the inside of the square root: x^2 + 4 must be ≥ 0, which is always true for all real x since x^2 ≥ 0. So the domain is all real numbers. The other forms would represent different operations (like adding x^2 separately or altering the inside of the square root), so they don’t match the composition here.

Composing means you feed g(x) into f. With f(u) = √(u + 4) and g(x) = x^2, you get f∘g(x) = √(g(x) + 4) = √(x^2 + 4). The domain is determined by the inside of the square root: x^2 + 4 must be ≥ 0, which is always true for all real x since x^2 ≥ 0. So the domain is all real numbers. The other forms would represent different operations (like adding x^2 separately or altering the inside of the square root), so they don’t match the composition here.