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

Which of the following functions are continuous for all real numbers? $\newline$ $\begin{array}{l} f(x)=\tan (x) \\ h(x)=x^{3} \end{array}$ $\newline$ Choose $1$ answer: $\newline$ (A) $f$ only $\newline$ (B) $h$ only $\newline$ (C) Both $f$ and $h$ $\newline$ (D) Neither $f$ nor $h$

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Math Problems

Algebra 1

Domain and range of quadratic functions: equations

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

\newline

\begin{array}{l} f(x)=\tan (x) \\ h(x)=x^{3} \end{array}

\newline

Choose

1

answer:

\newline

(A)

f

only

\newline

(B)

h

only

\newline

f

and

h

\newline

(D) Neither

f

nor

h

Understand Function Properties: To determine if the functions are continuous for all real numbers, we need to understand the properties of each function.
Analyze $f(x) = \tan(x)$ : Let's start with $f(x) = \tan(x)$ . The tangent function is the ratio of the sine function to the cosine function, $\tan(x) = \frac{\sin(x)}{\cos(x)}$ . We know that the cosine function has zeros at odd multiples of $\frac{\pi}{2}$ , which means that $\tan(x)$ will have vertical asymptotes at these points and will not be defined there. Therefore, $f(x) = \tan(x)$ is not continuous at these points.
Consider $h(x) = x^3$ : Now let's consider $h(x) = x^3$ . The function $x^3$ is a polynomial function, and polynomial functions are continuous everywhere. Therefore, $h(x) = x^3$ is continuous for all real numbers.
Final Comparison: Since $f(x) = \tan(x)$ is not continuous for all real numbers due to its vertical asymptotes, and $h(x) = x^3$ is continuous everywhere, the correct answer is that only $h(x) = x^3$ is continuous for all real numbers.