Question

a. Is it possible for a rational function to have both slant and horizontal asymptotes? Explain. b. Is it possible for an algebraic function to have two different slant asymptotes? Explain or give an example.

Short answer

Can an algebraic function have two different slant asymptotes? Answer: No, a rational function cannot have both slant and horizontal asymptotes. This is because they have contradictory degree requirements for the numerator and denominator polynomials. For slant asymptotes, the degree of the numerator should be one greater than the degree of the denominator, while for horizontal asymptotes, the degree of the numerator should be less than or equal to the degree of the denominator. Also, it is not possible for an algebraic function to have two different slant asymptotes, as it can only have one slant asymptote when the numerator has exactly one more degree of freedom than the denominator.

Step-by-step solution

a. Slant and horizontal asymptotes in rational functions

Rational functions are defined as the quotient of two polynomials, say P(x) and Q(x). Let's represent our rational function as R(x) = P(x) / Q(x). Slant asymptote (or oblique asymptote) occurs if the degree of P(x) is one more than the degree of Q(x). That is, if deg(P) = deg(Q) + 1. In this case, the slant asymptote is the result of dividing P(x) by Q(x) using polynomial long division, disregarding the remainder. Horizontal asymptote occurs if the degree of P(x) is less than or equal to the degree of Q(x). If deg(P) < deg(Q), R(x) approaches 0 as x approaches infinity, and thus the function has a horizontal asymptote at y = 0. If deg(P) = deg(Q), the function will have a horizontal asymptote at y = (leading coefficient of P) / (leading coefficient of Q). For a rational function to have both slant and horizontal asymptotes, its degree of P(x) should satisfy both conditions mentioned above, which is not possible. Therefore, a rational function cannot have both slant and horizontal asymptotes.

b. Two different slant asymptotes for an algebraic function

An algebraic function is a function built from a finite number of polynomials, using the arithmetic operations (+, -, ×, and ÷) and compositions (e.g., f(g(x))). For an algebraic function to have two different slant asymptotes, it must approach two different lines as x approaches positive and negative infinity. Let F(x) be an algebraic function. For this exercise, we will consider a specific type of algebraic function: rational functions. Recall that a rational function has a slant asymptote if deg(P) = deg(Q) + 1. Now, consider this rational function G(x) = (x^2 + x + 1) / (x - 1). The numerator has degree 2 and the denominator has degree 1, which meets the condition for a slant asymptote. However, if we simplify the function using polynomial long division, we get: G(x) = x + 2 + 3 / (x - 1) In this case, G(x) has one slant asymptote y = x + 2 and one vertical asymptote x = 1. However, when we try to come up with a more general algebraic function that can possess two different slant asymptotes, we run into a problem. If the algebraic function has exactly one more degree of freedom in the numerator than in the denominator, then it will only have one slant asymptote. If we increase the degree of freedom in the numerator beyond that, we no longer have a slant asymptote. Thus, it is not possible for an algebraic function to have two different slant asymptotes.

Key concepts

Slant Asymptote

A slant asymptote, also known as an oblique asymptote, appears in a rational function when the degree of the polynomial in the numerator is exactly one more than the degree of the polynomial in the denominator. This means if we represent our function as \[ R(x) = \frac{P(x)}{Q(x)} \]where \( \deg(P) = \deg(Q) + 1 \). To find the slant asymptote, you perform polynomial long division of \( P(x) \) by \( Q(x) \) and ignore the remainder. The quotient you get forms the equation of the slant asymptote.

• It's important to remember that a rational function can have only one slant asymptote.
• Slant asymptotes occur typically when \( \deg(P) = n+1 \) and \( \deg(Q) = n \), where \( n \) is a non-negative integer.

Understanding slant asymptotes is crucial because they tail off differently than horizontal asymptotes, usually representing a line that the graph of the function approaches as \( x \) moves towards positive or negative infinity.

Horizontal Asymptote

Horizontal asymptotes of rational functions are determined by comparing the highest degree terms of the numerator and the denominator. If we have a rational function\[ R(x) = \frac{P(x)}{Q(x)} \]then the degree of \( P(x) \) and \( Q(x) \) sets the stage for a horizontal asymptote. Here are the key scenarios:

• If \( \deg(P) < \deg(Q) \), the horizontal asymptote is \( y = 0 \).
• If \( \deg(P) = \deg(Q) \), the horizontal asymptote is given by \( y = \frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q} \).
• No horizontal asymptote exists if \( \deg(P) > \deg(Q) \).

Horizontal asymptotes provide a crucial understanding of the end behavior of a function, explaining how the function behaves as \( x \) grows very large or very small. Importantly, a rational function cannot have both a slant and a horizontal asymptote, as the conditions for each are mutually exclusive.

Algebraic Function Asymptotes

Algebraic functions are formed through a combination of basic arithmetic operations on polynomials. Asymptotes in such functions help us understand their behavior at extreme values, such as infinity. Primarily, we consider rational functions — the quotient of two polynomial expressions — when discussing asymptotes. For algebraic functions like rational functions, the rules determining asymptotes are:

• A function will have a vertical asymptote at points where the denominator is zero and the numerator isn’t, indicating undefined points that the function approaches but never touches.
• Horizontal and slant asymptotes follow the rules of degree comparisons between the numerator and denominator polynomials. Refer to the earlier sections for specifics on slant and horizontal asymptote conditions.

Algebraic functions generally do not have two distinct slant asymptotes due to the polynomial structure. The degrees of numerator and denominator essentially dictate a single asymptotic directional movement. This understanding helps in graphing and comprehending the behavior of these functions, especially where they diverge or taper at the borders of their graphs.