Question

function cubic function: f(x)=x^3
left end behavior (use limit notation):
horizontal asymptote(s): (write in the form of y = __ )
sketch of graph
right end behavior (use limit notation):
vertical asymptote(s): (write in the form of x = __ )
write using interval notation: intervals of increasing;
intervals of decreasing;
write using interval notation: domain;
write using interval notation: range;

Explanation:

Step1: Find derivative

The derivative of $f(x)=x^{3}$ is $f'(x) = 3x^{2}$.

Step2: Determine intervals of increase/decrease

Since $f'(x)=3x^{2}\geq0$ for all real - x, and $f'(x) = 0$ only at $x = 0$. The function is increasing on $(-\infty,\infty)$ and there is no interval of decrease.

Step3: Analyze end - behavior

For the left - end behavior, $\lim_{x
ightarrow-\infty}x^{3}=-\infty$. For the right - end behavior, $\lim_{x
ightarrow\infty}x^{3}=\infty$.

Step4: Find domain and range

The domain of a polynomial function $y = x^{3}$ is $(-\infty,\infty)$ because we can input any real number for x. The range is also $(-\infty,\infty)$ because for any real number y, we can find an x such that $x^{3}=y$.

Step5: Identify asymptotes

Since $y = x^{3}$ is a polynomial function, it has no horizontal asymptotes (because $\lim_{x
ightarrow\pm\infty}x^{3}=\pm\infty$) and no vertical asymptotes (no values of x that make the function undefined).

Answer:

• Intervals of Increasing: $(-\infty,\infty)$
• Intervals of Decreasing: None
• Left End Behavior: $\lim_{x

ightarrow-\infty}x^{3}=-\infty$

• Right End Behavior: $\lim_{x

ightarrow\infty}x^{3}=\infty$

• Domain: $(-\infty,\infty)$
• Horizontal Asymptote(s): None
• Vertical Asymptote(s): None
• Range: $(-\infty,\infty)$