# Analyze and sketch the graph of f(x)=\frac{3x}{x^2-1}.

## Question:

Analyze and sketch the graph of {eq}f(x)=\frac{3x}{x^2-1}{/eq}.

## Rational Functions

When we analyze rational functions or expression over the real numbers we have the issue of the domain.

All numbers that make the denominator zero have to be excluded. The behavior of the function about these

gaps in the domain has to be investigated. The function could tend to plus or minus infinity or just have a gap,

one point missing, in the graph. The other issue is when the variable x goes to plus or minus infinity. What

is the behavior of the function? Does it also tend to plus or minus infinity or does it approach a fixed value?

There may be local extreme points, but not always and points of inflection may occur also.

The very first item of the analysis is the maximal domain of {eq}f(x) {/eq} with respect to the

real numbers. That is, we have to exclude the...

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