Solution assignment 10 Fractional functions and graphs

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Assignment 10

Investigate the function:


and sketch its graph.


For x=0 we have a vertical asymptote (for x=0 the function becomes \pm\infty). When we try to find a horizontal asymptote we encounter some problems, because for x\to\infty or x\to-\infty the graph approaches \pm\infty. In order to get more information of the behavior of the function for x\to\infty or x\to-\infty, we rewrite the function as follows:

y=\displaystyle\frac{x^3-1}{x}=\displaystyle\frac{x^3}{x}-\displaystyle\frac{1}{x}= x^2-\displaystyle\frac{1}{x}

We see that the graph approaches:


when x\to\infty or x\to-\infty. In fact, this parabola is a special asmptote.
It is clear that the graph has no intersection points with the Y-axis (x=0). The intersection point with the X-axis can be found by solving the equation:


We notice immediately that this equation has the solution x=1 and thus we can factorize the function x^3-1:


The quadratic function has a discriminant which is less than 0 is and thus has no solutions. Thus the only intersection point with the X-as is (1,0).
Because of the special nature of the function we look at it in more detail near x=0. We notice that the second term \displaystyle\frac{1}{x} near x=0 has either the largest (positive or negative) value:

\displaystyle\lim_{x\to 0^{+}}=-\infty


\displaystyle\lim_{x\to 0^{-}}=+\infty

Based on this investigation we can sketch the graph. We also depicted the parabola.


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