Problem 91 - A Graphical Approach to Precalculus with Limits

Question

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{2 x^{2}}{x^{4}+1}$$

Step-by-Step Solution

Answer

Domain: all reals; Range: \([0, \infty)\); Horizontal asymptote at \(y=0\); Symmetric about the y-axis.

Step 1: Identify Domain

The function is \( f(x) = \frac{2x^2}{x^4 + 1} \). The denominator \( x^4 + 1 \) is never zero since \( x^4 + 1 \) is always positive. Therefore, the domain is all real numbers, i.e., \( (-\infty, \infty) \).

Step 2: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero. Since \( x^4 + 1 > 0 \) for all \( x \), there are no vertical asymptotes.

Step 3: Find the Horizontal Asymptotes

As \( x \to \infty \) or \( x \to -\infty \), the function behaves like \( \frac{2x^2}{x^4} = \frac{2}{x^2} \), which approaches 0. Therefore, there is a horizontal asymptote at \( y = 0 \).

Step 4: Check for Symmetry

To determine symmetry, check \( f(-x) = \frac{2(-x)^2}{(-x)^4 + 1} = \frac{2x^2}{x^4 + 1} \) which equals \( f(x) \). This indicates the function is even and symmetric about the y-axis.

Step 5: Determine the Range

Since the function is symmetric around the y-axis, with a minimum value at \( x=0 \) where \( f(0)=0 \), and \( f(x) \to 0 \) as \( x \to \pm \infty \), the range is \( [0, \infty) \).

Step 6: Plot the Graph

Plotting the function, we note the behavior: it is symmetric around the y-axis, increases from 0, reaches a maximum, and decreases towards 0 along the horizontal asymptote at \( y = 0 \).

Key Concepts

Understanding the Domain and Range of Rational Functions

The domain of a function encompasses all the possible input values (typically, x-values) that will not cause the function to be undefined. In rational functions like \( f(x) = \frac{2x^2}{x^4 + 1} \), we need to consider the denominator. Here, the denominator is \( x^4 + 1 \), which is always positive because adding 1 to any fourth power keeps it positive.

Thus, the domain is all real numbers, expressed as \( (-\infty, \infty) \).

• There are no exclusions due to division by zero or any other restrictions, making this function applicable for all x-values.

The range refers to the set of possible output values (y-values). For our function, its range is determined by how the function behaves for different x-values.

Since the lowest point occurs at \( x = 0 \) where \( f(0) = 0 \), and tends towards 0 as \( x \) moves to infinity, the range of this function is \( [0, \infty) \). The function may reach increasingly small values but never negative; thus, y-values stay non-negative.

Exploring Asymptotes in Rational Functions

Asymptotes are lines that a graph approaches but never touches. Common types include vertical and horizontal asymptotes. Let’s explore these for the function \( f(x) = \frac{2x^2}{x^4 + 1} \):

• Vertical Asymptotes: Vertical asymptotes occur where the denominator of a rational function equals zero, leading to undefined points. For our function, \( x^4 + 1 \) is never zero, which means there are no vertical asymptotes.
• Horizontal Asymptotes: These occur when \( x \) either tends to positive or negative infinity. For our function, as \( x \to \pm \infty \), the terms inside the fraction simplify to \( \frac{2}{x^2} \), which approaches 0. Hence, the horizontal asymptote here is \( y = 0 \).

Graphing Rational Functions By Hand

When graphing rational functions, having knowledge about its domain, range, and asymptotes aids in predicting its shape and behavior. Let's see how to do this for \( f(x) = \frac{2x^2}{x^4 + 1} \)

The graph is symmetric around the y-axis, showing even function behavior. Graphing by hand starts by determining critical points:

• The origin \( (0, 0) \) is a critical point since that's where the function achieves its minimum.
• Moving to more positive or negative x-values, the function increases, reaching a peak before leveling back towards and aligning near \( y = 0 \) due to the horizontal asymptote.

When plotting:

• Anchor points like the origin and understanding symmetry help delineate the general structure of the graph.
• The absence of x-values where the function becomes undefined ensures a smooth curve throughout its course.

Investigating Symmetry in Functions

Symmetry in functions helps in simplifying their analysis and graphing. There are various types of symmetry, and here we focus on even symmetry. A function is even if \( f(-x) = f(x) \) for all x in its domain.

For our function, \( f(x) = \frac{2x^2}{x^4 + 1} \):

• Substituting \(-x\) into the function gives \( f(-x) = \frac{2(-x)^2}{(-x)^4 + 1} = \frac{2x^2}{x^4 + 1} = f(x) \). Thus, it remains unchanged, confirming that this function is even.

Such symmetry implies that the graph is mirrored across the y-axis. This property simplifies graphing because knowing the behavior in one quadrant informs the behavior in the opposite quadrant.