Verified Answer :
- Determine the structure of the function \(f(x) = K \frac{x^2-16}{x-2}\) using the given \(x\)-intercepts and vertical asymptote.
- Perform polynomial division to express the function as \(f(x) = K(x+2) - \frac{12K}{x-2}\).
- Match the linear term to the given oblique asymptote \(y=x+2\) to find \(K=1\).
- Final rational function is \(\boxed{f(x) = \frac{x^2 - 16}{x - 2}}\).
Explanation
- Analyzing the Given Characteristics
To find the rational function \(f(x) = \frac{P(x)}{Q(x)}\) with the given characteristics, we first analyze the requirements. The \(x\)-intercepts are at \((-4, 0)\) and \((4, 0)\), which implies that the numerator \(P(x)\) must have roots at \(x = -4\) and \(x = 4\). Thus, \(P(x)\) can be represented as \(k(x + 4)(x - 4) = k(x^2 - 16)\) for some constant \(k\). The vertical asymptote is at \(x = 2\), which indicates that the denominator \(Q(x)\) must have a factor of \((x - 2)\). Since the oblique asymptote exists, the degree of the numerator must be exactly one higher than the degree of the denominator. Since the numerator is a quadratic (degree 2), the denominator must be linear (degree 1). Therefore, \(Q(x) = a(x - 2)\) for some constant \(a\). Combining these, the function takes the form \(f(x) = K \frac{x^2 - 16}{x - 2}\), where \(K = \frac{k}{a}\).
- Determining the Constant K and Formulating the Function
To determine the value of \(K\), we perform polynomial long division on the expression \(\frac{x^2 - 16}{x - 2}\). We rewrite the numerator as:
\[x^2 - 16 = (x^2 - 2x) + (2x - 4) - 12 = x(x - 2) + 2(x - 2) - 12 = (x + 2)(x - 2) - 12.\]
Substituting this back into the function:
\[f(x) = K \left( \frac{(x + 2)(x - 2) - 12}{x - 2} \right) = K \left( x + 2 - \frac{12}{x - 2} \right) = K(x + 2) - \frac{12K}{x - 2}.\]
As \(x \to \infty\), the term \(\frac{12K}{x - 2}\) approaches \(0\), so the function behaves like the linear function \(y = K(x + 2)\). We are given that the oblique asymptote is \(y = x + 2\), so we must have \(K = 1\).
- Final Verification and Conclusion
With \(K = 1\), the rational function is:
\[f(x) = \frac{x^2 - 16}{x - 2}.\]
We can verify the properties:
- \(x\)-intercepts: Setting \(f(x) = 0\) gives \(x^2 - 16 = 0\), which results in \(x = \pm 4\). The points are \((-4, 0)\) and \((4, 0)\).
- Vertical asymptote: The function is undefined at \(x = 2\), where the denominator is zero and the numerator is non-zero, confirming a vertical asymptote.
- Oblique asymptote: As shown through long division, \(f(x) = x + 2 - \frac{12}{x - 2}\), so as \(x \to \pm \infty\), \(f(x) \to x + 2\). The oblique asymptote is indeed \(y = x + 2\).
Thus, the rational function is \(\boxed{f(x) = \frac{x^2 - 16}{x - 2}}\).
Examples
In engineering and physics, rational functions with oblique asymptotes are often used to model systems that approach a linear trend as an independent variable (like time or distance) grows very large. For example, the trajectory of a projectile subject to air resistance or the voltage behavior in certain electrical circuits can be approximated by such functions. By identifying the asymptotes and intercepts, engineers can predict the long-term behavior of these systems, ensuring they remain within safe operational limits.