Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the rational function. Determine the domain of the function and identify any asymptotes. $$y=\frac{1+3 x^{2}-x^{3}}{x^{2}}$$

Step-by-Step Solution

Answer

The domain of the function is all real numbers except zero. The function tends to the line y=-x as \(x\) tends to infinity or minus infinity. There's a hole in the graph of the function at \(x=0\).

Step 1: Determine the domain

The domain of a function is all the values of x for which the function is defined. For a rational function, the denominator cannot be zero because division by zero is undefined. Setting the denominator \(x^{2}\) equal to zero and solving for x, the solution is \(x=0\). Thus, the function is undefined at \(x=0\), and the domain is all real numbers except for zero.

Step 2: Graph the function

To graph this rational function, you need to enter it on a graphing utility. This will yield a graphic representation of the function. Ensure that enough of the x- and y- axes are visible. The function will be undefined at \(x=0\), and you should see a hole in the graph at this value.

Step 3: Identify the asymptotes

An asymptote is a line that the graph of the function approaches as \(x\) tends to infinity or minus infinity. In this case, as the degree of the polynomial in the numerator is higher than the one in the denominator, research for oblique (slant) asymptotes. Divide \(1+3 x^{2}-x^{3}\) by \(x^{2}\) using polynomial division or a division algorithm. The result gives the equation of the slant asymptote. After division, get that the function behaves like -x as \(x\) tends to infinity or minus infinity.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we're referring to all the possible input values (usually represented by the variable \(x\)) for which the function is defined and giving real output values. With rational functions, things can get a bit tricky since they involve fractions. The golden rule for rational functions is that the denominator cannot be zero. If the denominator were zero, we would end up dividing by zero, which is undefined in math.
In the exercise given, the denominator is \(x^2\). To find the domain, set this denominator to zero and solve for \(x\). The equation \(x^2 = 0\) gives \(x = 0\). This tells us the function is undefined at \(x = 0\).
Therefore, the domain of this particular rational function is all real numbers except \(x = 0\). In simpler terms, any number you can think of is part of the domain, except zero.

Graphing Utilities

Graphing utilities are powerful tools that help us visualize functions on a coordinate plane. These can be physical graphing calculators or software applications, like Desmos or GeoGebra. They are incredibly handy for students to understand the behavior and shape of functions without manually plotting numerous points. For the rational function in the exercise, inputting it into a graphing utility will show you the curve that represents the function. The graph will illustrate how the function behaves, and most importantly, it will reveal where the function becomes undefined—in this case, at \(x = 0\).
Seeing is understanding: As you graph the function, you'll notice a "hole" or discontinuity at \(x = 0\). This is because the function is not defined at this point. Using graphing utilities helps students get an intuitive grasp of concepts such as domain and asymptotes in visual form.

Asymptotes

Asymptotes are lines that a graph approaches but never really touches or crosses. They're like invisible boundaries. There are different types of asymptotes. In rational functions, the most common ones are vertical, horizontal, and slant (or oblique) asymptotes. For the function in question, at \(x = 0\), there is a vertical asymptote because the function value becomes undefined there. This corresponds to the hole in the graph observed using graphing utilities. Now, because the degree of the polynomial in the numerator (\