Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

Using long division, you find that \(f(x)=\frac{x^{2}+1}{x+1}=x-1+\frac{2}{x+1} .\) What is the slant asymptote of the graph of \(f ?\)

Step-by-Step Solution

Answer

The slant asymptote of the graph of \(f(x)\) is \(y = x - 1\).

Step 1: Identify function

Identify the given function which is \(f(x)=\frac{x^{2}+1}{x+1}\) from the exercise.

Step 2: Use long division

Using long division or polynomial division, divide \(x^{2}+1\) by \(x+1\) to get a quotient and a remainder. It is given in the problem that the quotient is \(x-1\) and remainder is \(\frac{2}{x+1}\).

Step 3: Identify the slant asymptote

The slant asymptote of a rational function is the quotient of the division. Ignore the remainder if the degree of the numerator is exactly one more than the degree of the denominator. In this case, ignoring the remainder of \(\frac{2}{x+1}\), the quotient \(x-1\) stands as the slant asymptote of the rational function \(f(x)\).

Key Concepts

Long Division

Long division is a method used to divide one polynomial by another. It works similarly to numerical long division.
You begin by dividing the highest degree term of the dividend with the highest degree term of the divisor. Then multiply the entire divisor by this quotient term and subtract the result from the original dividend.
Repeat the process with the new dividend, each time reducing the degree of the dividend until you cannot divide further.

• This method allows you to express a rational function as a sum of a polynomial and a remainder.
• For example, dividing the original polynomial \(x^2 + 1\) by \(x + 1\), we start by dividing \(x^2\) by \(x\), resulting in \(x\).
• We then multiply \(x + 1\) by \(x\) and subtract from \(x^2 + 1\).
• The result is a new polynomial which is \(0\cdot x + 1\), allowing the process to continue till completion.

By following these steps, the original problem gives the quotient \(x - 1\), which is crucial when identifying slant asymptotes.

Rational Function

A rational function is a function that can be expressed as the quotient of two polynomials. In mathematical terms, it is of the form \(f(x) = \frac{p(x)}{q(x)}\) where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) eq 0\).
Rational functions can have asymptotes, which are lines that the graph of the function approaches but never actually touches.

• There are different types of asymptotes: vertical, horizontal, and slant.
• A slant or oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.

In the given exercise, the rational function is \( f(x) = \frac{x^2 + 1}{x + 1} \). Here, the degree of the numerator is one more than the degree of the denominator, namely 2 versus 1, which suggests a slant asymptote may exist. Recognizing this relationship helps guide us in understanding the function’s behavior at extreme values of \(x\).

Polynomial Division

Polynomial division encompasses methods like long division to simplify expressions where polynomials are divided by other polynomials.
It is particularly useful in understanding the behavior of rational functions, especially in finding asymptotes.

• Just as numerical long division helps simplify complex arithmetic, polynomial division achieves similar simplifications in algebraic terms.
• In the method, each step reduces the complexity of the polynomial till it is simplified to a degree where division inconsequentially is impractical.
• In this example, with \(f(x) = \frac{x^2 + 1}{x + 1}\), polynomial division helps decompose \(f(x)\) into a polynomial part \(x - 1\) and a remainder part \(\frac{2}{x + 1}\).

The polynomial part \(x - 1\) not only simplifies the expression but also provides critical insight into asymptotic behavior, offering the slant asymptote directly from this division. This methodical approach assists in the understanding of rational functions and their corresponding graphs.