Given f(x) = x^2 + 1 and g(x) = sqrt(x), what is the domain of (g o f)(x)?

Question

Accepted Answer

All real numbers. The composition (g o f)(x) = g(f(x)) = sqrt(x^2 + 1). For this to be defined, we need x^2 + 1 >= 0 (the argument of the square root must be non-negative). Since x^2 >= 0 for all real numbers, x^2 + 1 >= 1 > 0 for all real numbers. Therefore the domain is all real numbers.

"x >= 0" (x >= 0) might be chosen by students who think x must be non-negative for a square root, but that restriction applies to sqrt(x), not sqrt(x^2 + 1). The expression inside the square root is x^2 + 1, which is always positive regardless of the sign of x. Even when x is negative, say x = -5, we get sqrt((-5)^2 + 1) = sqrt(26), which is perfectly defined.

Domains of composite functions require careful analysis on the CLEP exam. The domain of (g o f)(x) consists of all x in the domain of f such that f(x) is in the domain of g. Here, f(x) = x^2 + 1 is defined for all real numbers, and g requires non-negative inputs. Since f(x) = x^2 + 1 >= 1 > 0 always, every output of f is in the domain of g. This problem illustrates how composition can have a larger effective domain than the outer function alone.

CLEP College Algebra: Question of the Day

Get a free CLEP College Algebra question every day

Ready for the full CLEP College Algebra? Earn 3 college credits.