Problem 3 - Algebra and Trigonometry Real Mathematics, Real People

Question

Does the graph of \(f(x)=\frac{x^{3}-1}{x^{2}+2}\) have a slant asymptote?

Step-by-Step Solution

Answer

Yes, the function \(f(x)=\frac{x^{3}-1}{x^{2}+2}\) has a slant asymptote. The equation of the slant asymptote is \(y=x\).

Step 1: Determining the Degree of the Polynomials

To determine if the function has a slant asymptote, first check the degrees of the numerator and denominator. For the given function \(f(x)=\frac{x^{3}-1}{x^{2}+2}\), degree of the polynomial in the numerator (which is \(x^{3}-1\)) is 3 and that in the denominator (which is \(x^{2}+2)\) is 2.

Step 2: Long Division of Polynomials

Since the degree of the polynomial in the numerator is one more than the degree of the polynomial in the denominator, perform long division to find the equation of the slant asymptote. Upon performing the division of polynomials \(x^{3}-1\) by \(x^{2}+2\), the result is \(x\).

Step 3: Conclusion

The graph of the function \(f(x)=\frac{x^{3}-1}{x^2+2}\) approaches the line \(y=x\) as \(x\) approaches \(-\infty\) or \(+\infty\). Therefore, the function \(f(x)\) has a slant asymptote, and the equation of the slant asymptote is \(y=x\).

Key Concepts

Polynomial Long Division

Understanding polynomial long division is crucial for analyzing the behavior of rational functions, particularly when searching for slant asymptotes. Polynomial long division is analogous to the long division process you learned with numbers but is used for dividing one polynomial by another. Here's a simple guide to perform this operation:

First, write down the dividend (the polynomial you're dividing) and the divisor (the polynomial you're dividing by). Just like with numerical division, you divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. Multiply this term by the divisor, subtract the result from the dividend, and bring down the next term.

Continue the process until you bring down all terms of the dividend or the remainder has a degree less than the divisor. If the degree of the remainder is smaller than the degree of the divisor, the remainder can be expressed as the numerator of a fraction with the divisor as the denominator, which can contribute to the behavior of the rational function around the asymptote.

Degrees of Polynomials

The degree of a polynomial is determined by the highest power of the variable in the polynomial. For example, in the polynomial function f(x) = x^3 - 1, the highest power of x is 3, hence it's said to be a third-degree polynomial. Similarly, for the denominator x^2 + 2, the highest power is 2, making it a second-degree polynomial.

Understanding the degrees of polynomials involved in a rational function helps us forecast the function's behavior at extreme values of x. If the degree of the polynomial in the numerator is one higher than that of the polynomial in the denominator, as seen in our exercise, the rational function will have a slant (or oblique) asymptote. This asymptotic behavior can be found by using polynomial long division, which allows us to visualize how the function behaves as we move further out on the x-axis.

Rational Functions

Rational functions are fractions where both the numerator and denominator are polynomials. The function f(x) = (x^3 - 1) / (x^2 + 2) is a typical example of a rational function. These functions are interesting as they can exhibit various behaviors such as asymptotes, which are lines that the graph of the function approaches but never touches.

To fully grasp the characteristics of a rational function, you should analyze the degrees of its polynomials. Identifying horizontal asymptotes, vertical asymptotes, and slant asymptotes becomes more straightforward with this knowledge. Moreover, understanding the end behavior of these functions helps in sketching the rough graph and predicting how the function behaves for very large or very small values of x.

Asymptotic Behavior

The asymptotic behavior of a function refers to how the function behaves as the input or output approaches a certain value or infinity. For rational functions, asymptotes can be vertical, horizontal, or slant. Vertical asymptotes occur where the function is undefined, typically where the denominator is zero. Horizontal asymptotes are found when the degree of the polynomial in the numerator is less than or equal to the degree in the denominator, and it shows the output value that the function approaches as x tends to infinity.

In contrast, slant asymptotes occur when the degree of the numerator is exactly one more than the denominator, indicating that the end behavior involves the function approaching a line that isn't horizontal. In our exercise, f(x) = (x^3 - 1) / (x^2 + 2) approaches the line y = x as x goes to negative or positive infinity, which means it has a slant asymptote at y = x.