What is the domain of a real-valued function f?

• A. The set of all real numbers x for which f(x) is a real number.
• B. The set of all possible outputs y of f.
• C. The derivative of f.
• D. The range of f.

Answer: (A)

Understanding what domain means helps here. The domain of a real-valued function is the set of input values x for which f(x) is defined and yields a real number. In other words, it’s all the x-values you can plug into the function so that the output is a real number. That’s exactly what the option describing “the set of all real numbers x for which f(x) is a real number” says.

Think of examples to ground the idea: for f(x) = sqrt(x−1), you can only plug in x ≥ 1, so the domain is [1, ∞). For f(x) = 1/x, you must avoid x = 0, so the domain is all real numbers except 0. The other terms refer to different ideas: the range is the set of possible outputs y, the derivative is a rate of change, not a set of inputs.