Answered

What is the equation of the slant asymptote of the rational function?

$f(x)=\frac{10 x^3-15 x^2+x-1}{}$

Verified Answer :

  • Analyze the rational function to confirm the degree of the numerator is one higher than the degree of the denominator.
  • Perform polynomial long division of \(10x^3 - 15x^2 + x - 1\) by \(x^2 + 1\).
  • Recognize that as \(x \to \pm\infty\), the fractional remainder \(\frac{-9x + 14}{x^2 + 1}\) vanishes to 0.
  • State the final linear quotient as the slant asymptote: \(\boxed{y = 10x - 15}\).

Explanation

  1. Problem Analysis and Condition for Slant Asymptote

To find the slant asymptote of the rational function \(f(x) = \frac{10x^3 - 15x^2 + x - 1}{x^2 + 1}\), we look for a linear function \(y = mx + b\) that the graph approaches as \(x\) approaches positive or negative infinity. A rational function has a slant asymptote if the degree of the numerator is exactly one higher than the degree of the denominator. Here, the numerator has degree 3 and the denominator has degree 2, confirming that a slant asymptote exists.

  1. Performing Polynomial Long Division

To find the slant asymptote, we perform polynomial long division of the numerator \(10x^3 - 15x^2 + x - 1\) by the denominator \(x^2 + 1\).

  1. Divide the leading term of the numerator by the leading term of the denominator: \(\frac{10x^3}{x^2} = 10x\).
  2. Multiply \(10x\) by the divisor: \(10x(x^2 + 1) = 10x^3 + 10x\).
  3. Subtract this from the original numerator: \((10x^3 - 15x^2 + x - 1) - (10x^3 + 10x) = -15x^2 - 9x - 1\).
  4. Divide the leading term of the result by the leading term of the divisor: \(\frac{-15x^2}{x^2} = -15\).
  5. Multiply \(-15\) by the divisor: \(-15(x^2 + 1) = -15x^2 - 15\).
  6. Subtract this from the previous result: \((-15x^2 - 9x - 1) - (-15x^2 - 15) = -9x + 14\).

This results in the expression \(f(x) = 10x - 15 + \frac{-9x + 14}{x^2 + 1}\).

  1. Identifying the Slant Asymptote

As \(x\) approaches positive or negative infinity, the remainder term \(\frac{-9x + 14}{x^2 + 1}\) approaches 0 because the degree of the denominator is higher than the degree of the numerator in the remainder. Therefore, the function \(f(x)\) approaches the linear function \(y = 10x - 15\). This is the slant asymptote of the rational function.

  1. Final Answer

The slant asymptote of the given rational function is \(\boxed{y = 10x - 15}\).

Examples

Slant asymptotes are useful in analyzing the long-term behavior of complex systems. For instance, if an industrial cooling process has a temperature variation represented by a rational function, finding the slant asymptote allows engineers to predict the long-term steady-state trend of the temperature as time increases, even if the system has initial fluctuations. Understanding this behavior is critical for maintaining equipment stability and efficiency.

Related Questions