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Which of the following functions are continuous for all real numbers? {:[g(x)=root(5)(x)],[h(x)=root(3)(x)]:} Choose 1 answer: (A) g only (B) h only (C) Both g and h (D) Neither g nor h

Which of the following functions are continuous for all real numbers?\newlineg(x)=x5h(x)=x3 \begin{array}{l} g(x)=\sqrt[5]{x} \\ h(x)=\sqrt[3]{x} \end{array} \newlineChoose 11 answer:\newline(A) g g only\newline(B) h h only\newline(C) Both g g and h h \newline(D) Neither g g nor h h

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Full solution

Q. Which of the following functions are continuous for all real numbers?\newlineg(x)=x5h(x)=x3 \begin{array}{l} g(x)=\sqrt[5]{x} \\ h(x)=\sqrt[3]{x} \end{array} \newlineChoose 11 answer:\newline(A) g g only\newline(B) h h only\newline(C) Both g g and h h \newline(D) Neither g g nor h h
  1. Analyzing g(x)g(x): Let's analyze the first function:\newlineg(x)=x5g(x) = \sqrt[5]{x}\newlineThis is the fifth root of xx. The fifth root function is defined for all real numbers because we can take the fifth root of any real number, whether it is positive, negative, or zero.
  2. Analyzing h(x)h(x): Now let's analyze the second function:\newlineh(x)=x3h(x) = \sqrt[3]{x}\newlineThis is the cube root of xx. The cube root function is also defined for all real numbers because we can take the cube root of any real number, just like the fifth root.
  3. Continuity of g(x)g(x) and h(x)h(x): Since both g(x)g(x) and h(x)h(x) are defined for all real numbers and do not have any points of discontinuity, we can conclude that both functions are continuous for all real numbers.
  4. Conclusion: Therefore, the correct answer is:\newline(C)(C) Both gg and hh

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