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Which of the following functions are continuous for all real numbers?

h(x)=log(x)

g(x)=cot(x)
Choose 1 answer:
(A)
h only
(B)
g only
(C) Both
h and
g
(D) Neither
h nor
g

Which of the following functions are continuous for all real numbers? $\newline$ $h(x)=\log (x)$ $\newline$ $g(x)=\cot (x)$ $\newline$ Choose $1$ answer: $\newline$ (A) $h$ only $\newline$ (B) $g$ only $\newline$ (C) Both $h$ and $g$ $\newline$ (D) Neither $h$ nor $g$

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Identify properties of logarithms

Full solution

Q. Which of the following functions are continuous for all real numbers?

\newline

h(x)=\log (x)

\newline

g(x)=\cot (x)

\newline

Choose

1

answer:

\newline

(A)

h

only

\newline

(B)

g

only

\newline

(C) Both

h

and

g

\newline

(D) Neither

h

nor

g

Domain of $h(x)$ : To determine if $h(x) = \log(x)$ is continuous for all real numbers, we need to consider the domain of the logarithmic function. The logarithmic function is defined only for positive real numbers. Therefore, $h(x)$ is not continuous for all real numbers because it is not defined for $x \leq 0$ .
Cotangent function: Now, let's consider $g(x) = \cot(x)$ . The cotangent function is the reciprocal of the tangent function. It is undefined whenever the tangent function is zero, which occurs at integer multiples of $\pi$ . Therefore, $g(x)$ is not continuous at these points and is not continuous for all real numbers.
Conclusion: Since neither $h(x) = \log(x)$ nor $g(x) = \cot(x)$ is continuous for all real numbers, the correct answer is $(D)$ Neither $h$ nor $g$ .

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