Qualitative graph for a function of $x^{3}$.

Question

I have this function: $$f(x)=\frac{x(x^2-3)}{x^3-3x-3}$$ I need to draw its graph. I've tried with a classic study of a function but it cames out a mess. Any idea to simplify the study for draw the graph? Thank you.

Answer 1

HINT

Note that

$$f(x)=\frac{x(x^2-3)}{x^3-3x-3}=f(x)=\frac{x^3-3x}{x^3-3x-3}=f(x)=1+\frac{3}{x^3-3x-3}$$

For a first sketch

• determine domain
• find the value for some "special" and/or "simple" point as x=0,1,etc.
• find the value for which denominator = 0 (and thus vertical asymptothes)
• find limit at $\pm \infty$

Then for a complete study we need use derivatives.

Answer 2

In order to get a graph of $$f(x)=\frac{x(x^2-3)}{x^3-3x-3}$$

We look at intercepts, asymptotes and a few more points.

$f(x)=0$ gives you your $x$-intercepts.

$$f(x)=0 \implies x(x^2-3)=0 \implies x=0,\pm \sqrt 3$$ $$ x^3-3x-3=0 $$ gives you vertical asymptotes $$ x^3-3x-3=0\implies x=2.1038$$

which is the only vertical asymptote.

$$ \lim _{x\to \pm \infty }f(x) = 1$$

That is $y=1$ is a horizontal asymptote.

We use the above information to sketch a graph.

Then we verify our graph with a graphing utility.