Problem 95 - A Graphical Approach to College Algebra

Question

Becomes $$\begin{aligned}&f(x)=\frac{x^{5}+x^{4}+x^{2}+1}{x^{4}+1}\\\&f(x)=x+1+\frac{x^{2}-x}{x^{4}+1} \end{aligned}$$ after the numerator is divided by the denominator. (a) What is the equation of the oblique asymptote of the graph of the function? (b) For what \(x\) -value(s) does the graph of the function intersect its asymptote? (c) As \(x \rightarrow \infty,\) does the graph of the function approach its asymptote from above or below?

Step-by-Step Solution

Answer

(a) \( y = x + 1 \); (b) \( x = 0, 1 \); (c) From above.

Step 1: Perform Polynomial Long Division

Divide the numerator \( x^5 + x^4 + x^2 + 1 \) by the denominator \( x^4 + 1 \) using polynomial long division.1. Divide the first term of the numerator by the first term of the denominator: \( x^5 / x^4 = x \).2. Multiply the entire divisor \( x^4 + 1 \) by \( x \): \( (x)(x^4 + 1) = x^5 + x \).3. Subtract the result from the original numerator: \[ (x^5 + x^4 + x^2 + 1) - (x^5 + x) = x^4 + x^2 + 1 - x. \]4. Simplify: \( x^4 + x^2 - x + 1 \).5. Repeat these steps using the result \( x^4 + x^2 - x + 1 \) divided by \( x^4 + 1 \): \[ x^4 / x^4 = 1 \]6. Multiply and subtract as before: \[ (x^4 + x^2 - x + 1) - (1)(x^4 + 1) = x^2 - x \]This gives us the quotient \( x + 1 \) and remainder \( x^2 - x \).

Step 2: Write the Function in Quotient Form

After dividing, we find:\[ f(x) = x + 1 + \frac{x^2 - x}{x^4 + 1}\]This shows the function expressed as the quotient plus a remainder.

Step 3: Identify the Oblique Asymptote

For rational functions, the oblique asymptote is given by the quotient from polynomial long division. Here, the oblique asymptote is \( y = x + 1 \).

Step 4: Find Intersection Points

Set the function equal to its asymptote to find the intersection points:\[x + 1 + \frac{x^2 - x}{x^4 + 1} = x + 1\]Simplifying:\[\frac{x^2 - x}{x^4 + 1} = 0\]This occurs when \( x^2 - x = 0 \), solving gives:\[x(x - 1) = 0\]So, the graph intersects the asymptote at \( x = 0 \) and \( x = 1 \).

Step 5: Determine the Behavior at Infinity

Consider the leading terms of the remainder \( \frac{x^2 - x}{x^4 + 1} \) as \( x \rightarrow \infty \). As the degree of the numerator is less than the degree of the denominator, \( \frac{x^2 - x}{x^4 + 1} \rightarrow 0 \).Since \( \frac{x^2 - x}{x^4 + 1} \) approaches zero from the positive values for large \( x \), the function approaches its asymptote from above as \( x \rightarrow \infty \).

Key Concepts

Polynomial Long Division

Polynomial long division is a process similar to numerical long division. It helps us find how one polynomial divides another, yielding a quotient and a remainder. This technique is essential when dealing with rational functions, particularly when simplifying or rewriting them.

• Start by dividing the leading term of the numerator by the leading term of the denominator. In our example, the first term of the numerator is \( x^5 \), and the first term of the denominator is \( x^4 \), which gives us \( x \) as the first part of our quotient.
• Multiply this result by the entire divisor and subtract from the original polynomial. It changes the problem by removing terms, focusing on the remainder that needs further division.
• Repeat this process with the new polynomial formed after subtraction until the remainder's degree is less than the divisor.

This method enabled us to rewrite the function \( f(x) = \frac{x^{5}+x^{4}+x^{2}+1}{x^{4}+1} \) as \( x + 1 + \frac{x^2 - x}{x^{4} + 1} \). The quotient \( x + 1 \) is particularly important, as it represents our function's oblique asymptote.

Rational Functions

Rational functions are expressions represented as the ratio of two polynomials. They are relatively common in mathematics and often involve complex behavior compared to simple linear or polynomial functions.

• The general form is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).
• Key characteristics of rational functions include vertical asymptotes, horizontal asymptotes, and oblique asymptotes, as seen in our function.
• Vertical asymptotes occur where the denominator \( Q(x) \) is zero, but there is no corresponding zero in the numerator \( P(x) \) at that point.
• Oblique (or slant) asymptotes happen when the degree of the numerator is one more than the degree of the denominator, as illustrated in our function, with the asymptote \( y = x + 1 \).

Understanding these characteristics helps us graph rational functions effectively, offering insights into possible intersections, end behavior, and the influence of roots.

Graph Intersection

The graph of a function and its asymptote can intersect in points where the remainder of the division equals zero. This reveals shared solutions between function and asymptote, marking actual touchpoints on a graph.

• To identify these intersection points, set the remainder part of the function equal to zero. For \( f(x) = x + 1 + \frac{x^2 - x}{x^4 + 1} \), equate \( \frac{x^2 - x}{x^4 + 1} \) to zero.
• Solve the numerator of the remainder \( x^2 - x = 0 \) to find \( x(x - 1) = 0 \). It gives the solutions \( x = 0 \) and \( x = 1 \).
• Thus, the graph intersects the oblique asymptote at these values, precisely where the function simplifies to just its asymptote.

These intersections can be crucial when examining behavioral changes in graphs, helping predict how the function behaves in proximity to these lines as \( x \) approaches these specific values.