Problem 44 - Calculus

Question

Which of the functions I-III meet each of the following descriptions? There may be more than one function for each description, or none at all. (a) Horizontal asymptote of \(y=1\) (b) The \(x\) -axis is a horizontal asymptote. (c) Symmetric about the \(y\) -axis. (d) An odd function. (e) Vertical asymptotes at \(x=\pm 1\) I. \(y=\frac{x-1}{x^{2}+1}\) II. \(y=\frac{x^{2}-1}{x^{2}+1}\) III. \(y=\frac{x^{2}+1}{x^{2}-1}\)

Step-by-Step Solution

Answer

(a) II and III; (b) I; (c) II, III; (d) I; (e) III.

Step 1 - Identify Horizontal Asymptotes

**Function I:** As the degree of the denominator is higher than the numerator, the horizontal asymptote is at \( y = 0 \).**Function II:** The degrees of the numerator and denominator are equal, so the horizontal asymptote is the ratio of the leading coefficients: \( \frac{1}{1} = 1 \).**Function III:** The degrees of the numerator and denominator are equal, so the horizontal asymptote is the ratio of the leading coefficients: \( \frac{1}{1} = 1 \).

Step 2 - Check if x-axis is a Horizontal Asymptote

The horizontal asymptote is the \(x\)-axis (i.e., \(y = 0\)) only for those functions where the degree of the denominator is greater than the numerator. **Function I:** Satisfies this with a horizontal asymptote of \(y = 0\).**Function II & III:** Both have horizontal asymptotes other than \(y = 0\).

Step 3 - Test Symmetry About the y-axis

Evaluate if replacing \( x \) with \( -x \) results in the same function, indicating even symmetry.**Function I:** \(y(x) = y(-x)\), as both have symmetrical outcomes with opposite signs, it’s not symmetric about the y-axis.**Function II:** \( y(-x) = \frac{(-x)^2 - 1}{(-x)^2 + 1} = \frac{x^2 - 1}{x^2 + 1} \), proving symmetry about the y-axis.**Function III:** Similar to II, it doesn’t change and remains the same, showing symmetry about the y-axis.

Step 4 - Evaluate Odd Function Characteristic

Determine if replacing \( x \) with \( -x \) gives the negative of the original function.**Function I:** \(y(-x) = -y(x)\), indicating it is an odd function.**Function II & III:** Both functions are not odd as their transformations with \(-x\) do not result in \(-y(x)\).

Step 5 - Determine Vertical Asymptotes

Identifying where the denominator equals zero but not the numerator, leading to a vertical asymptote.**Function I & II:** No vertical asymptotes as \(x^{2} + 1\) never equals zero (no real solutions for \(-1\)).**Function III:** The denominator \(x^{2} - 1 = (x-1)(x+1)\) leads to vertical asymptotes at \(x = 1\) and \(x = -1\).

Key Concepts

Rational functions

Rational functions are a core concept in algebra and calculus. They are functions represented as the ratio of two polynomials, such as \( y = \frac{P(x)}{Q(x)} \). The behavior of rational functions largely depends on the degrees of these polynomials:

• When the degree of the numerator is less than the degree of the denominator, the function typically approaches zero as \( x \to \pm \infty \).
• If the degrees are equal, the horizontal asymptote can be found by evaluating the ratio of the leading coefficients.
• When the numerator's degree is higher, there might not be a horizontal asymptote.

Rational functions like those we assessed in the exercise can exhibit various types of asymptotic behavior, informing us about the values the function tends to approach in the limits of large or small \( x \) values.

Symmetry in functions

Symmetry in functions helps identify function behavior through transformations, providing insights for graphing and analysis. A function is symmetric about the \( y \)-axis, considered an even function, if replacing \( x \) with \( -x \) results in the original function, \( f(-x) = f(x) \). This characteristic was found in Function II of our exercise.

Alternatively, a function may also be symmetric about the origin, known as odd symmetry, if replacing \( x \) with \( -x \) yields the negative of the function, \( f(-x) = -f(x) \). Function I from the exercise demonstrated this property, making it an example of odd function behavior. These properties of symmetry are crucial for quickly determining graphical properties and simplifying function analysis.

Odd and even functions

Odd and even functions are particularly useful concepts in mathematics, especially when analyzing their graphs and predicting behavior:

• Even functions are known for symmetry about the \( y \)-axis. Their graphs exhibit mirror-image behavior on either side of the \( y \)-axis, helping predict behavior without fully plotting the graph.
• Odd functions boast symmetry about the origin. Their behavior can be visualized by a 180-degree rotation around the origin, giving insights into their function values without complete plotting.

These characteristics not only inform us on symmetry but also dictate that even functions typically yield the same result at equal distances from the \( y \)-axis, while odd functions yield opposite results for opposite inputs. Recognizing these patterns helps simplify mathematical calculations and graphical estimates.

Graphical behavior of functions

Graphical behavior of functions such as rational functions reveals significant insights into their limits, asymptotes, and symmetry characteristics:

• Horizontal asymptotes provide insight into the end behavior of the graphs, where they approach a specific value. This was observed in Function II and III from our exercise with \( y = 1 \) as the asymptote.
• Vertical asymptotes indicate points where the function approaches infinity, typically due to the denominator in the function equating to zero. For instance, Function III showed vertical asymptotes at \( x = 1 \) and \( x = -1 \).
• Symmetrical behavior and characteristics, such as odd and even properties, can provide additional insights into functions' behavior over specific intervals, helping predict outcome trends over larger domains.

Understanding these aspects offers a clearer, simplified determination of function behavior which aids significantly in graph sketching and analysis.