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08 Oct 2025, 11:52
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Difficulty:
65%
(hard)
Question Stats:
60% (01:36) correct
40% (01:15) wrong based on 5 sessions
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Which of the following functions has as its domain the set of all real numbers?
Indicate all that apply.
A. \(f(x) = \frac{4}{x+4}\)
B. \(f(x) = \frac{4}{x^2+1}\)
C. \(f(x) = \sqrt{4x^3}\)
D. \(f(x) = \sqrt{(-4x)^2}\)
E. \(f(x) = \sqrt[3]{4x}\)
Indicate all that apply.
A. \(f(x) = \frac{4}{x+4}\)
B. \(f(x) = \frac{4}{x^2+1}\)
C. \(f(x) = \sqrt{4x^3}\)
D. \(f(x) = \sqrt{(-4x)^2}\)
E. \(f(x) = \sqrt[3]{4x}\)
11 Oct 2025, 06:50
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Official Explanation
The domain of a function, f(x), is the set of all possible x-values, or input values, for which the function is defined.
(A) is not correct, because if \(x = –4,\) the denominator is 0, and division by 0 is undefined.
(B) is correct, because the denominator \(x^2 + 1\) is always greater than zero, so no division by zero error occurs.
(C) is not correct, because if \(x < 0,\) then \(4x^3 < 0.\) This creates the square root of a negative, which is undefined within the real number system.
(D) is correct, because squaring the quantity (–4x) yields a value ≥ 0, and the square-root function is defined for all numbers ≥ 0.
(E) is correct, because the cube-root function is defined for all real numbers, including negatives.
Answer is (B), (D), and (E).
The domain of a function, f(x), is the set of all possible x-values, or input values, for which the function is defined.
(A) is not correct, because if \(x = –4,\) the denominator is 0, and division by 0 is undefined.
(B) is correct, because the denominator \(x^2 + 1\) is always greater than zero, so no division by zero error occurs.
(C) is not correct, because if \(x < 0,\) then \(4x^3 < 0.\) This creates the square root of a negative, which is undefined within the real number system.
(D) is correct, because squaring the quantity (–4x) yields a value ≥ 0, and the square-root function is defined for all numbers ≥ 0.
(E) is correct, because the cube-root function is defined for all real numbers, including negatives.
Answer is (B), (D), and (E).
