Problem 92 - 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: \((-\infty, \infty)\); Range: \((-\infty, 0]\); Horizontal asymptote at \(y=0\); Symmetric about y-axis.

Step 1: Domain Determination

To find the domain of the function \(f(x) = \frac{-2x^2}{x^4 + 1}\), identify values of \(x\) that do not make the denominator zero. Since \(x^4 + 1\) is always greater than zero for all real \(x\), the domain is all real numbers, or \((-\infty, \infty)\).

Step 2: Range Determination

As the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the graph will approach the x-axis but never touch or cross it, implying that the range will be below the x-axis. The range of \(f(x)\) is \((-\infty, 0]\).

Step 3: Vertical Asymptote Analysis

Vertical asymptotes occur where the denominator is zero. Since \(x^4 + 1\) is never zero, there are no vertical asymptotes for this function.

Step 4: Horizontal Asymptote Analysis

The horizontal asymptote of a rational function is found by comparing the degrees of the polynomials. Since the degree of the numerator \(-2x^2\) is less than the degree of the denominator \(x^4 + 1\), the horizontal asymptote is at \(y = 0\).

Step 5: Symmetry Analysis

Check for symmetry by testing if \(f(-x) = f(x)\) (even symmetry) or \(f(-x) = -f(x)\) (odd symmetry). Calculate \(f(-x) = \frac{-2(-x)^2}{(-x)^4 + 1} = \frac{-2x^2}{x^4 + 1} = f(x)\). Thus, the function is even and symmetric about the y-axis.

Step 6: Graphing the Function

Begin by marking the x-axis as the horizontal asymptote. Plot several points across different values of \(x\) to observe the function's behavior (e.g., \(f(0) = 0\), \(f(1) = \frac{-2}{2} = -1\), \(f(-1) = \frac{-2}{2} = -1\)). Since the function is even, the left side of the graph mirrors the right side.

Key Concepts

Domain and Range

When exploring rational functions, a crucial step is determining the domain and range. For the function given, \(f(x) = \frac{-2x^2}{x^4 + 1}\), the domain comprises all real numbers, expressed as \((-fty, fty)\). This is because the denominator, \(x^4 + 1\), remains positive for all \(x\) values, meaning it can never equal zero.
Understanding the range requires examining behavior as the x-values change. Here, the numerator's degree is less than the denominator's, resulting in the graph nearing but not crossing the x-axis. As such, the range is all real values below and including zero: \((-fty, 0]\). This indicates that the function's values are never positive.

Asymptotes

Asymptotes serve as invisible boundaries that guide a function's graph. With the given function, identifying vertical asymptotes involves locating points where the denominator equals zero. Since \(x^4 + 1\) is always positive, no vertical asymptotes exist for this function.
Horizontal asymptotes occur when examining the degrees of the numerator and denominator. In \(f(x) = \frac{-2x^2}{x^4 + 1}\), the numerator's polynomial degree of 2 is less than the denominator's degree of 4. This suggests the graph approaches the line \(y = 0\) as \(x\) moves towards infinity in either direction.

• No vertical asymptotes
• Horizontal asymptote at \(y = 0\)

Graph Symmetry

Graph symmetry can reveal much about the structure of a function. It can be determined by testing if the function replicates across either axis. For even symmetry, we test if \(f(-x) = f(x)\). In the equation \(f(x) = \frac{-2x^2}{x^4 + 1}\), substituting \(-x\) into the function yields the same result: \(\frac{-2x^2}{x^4 + 1}\).
This shows the function displays even symmetry, meaning the graph is symmetric about the y-axis. This type of symmetry simplifies graphing, as only half of the function needs to be plotted, with the other half mirroring it across the y-axis.

• Even symmetry across the y-axis

Polynomial Degree Comparison

The degrees of the polynomials in a rational function impact its end behavior and asymptotes. For the function \(f(x) = \frac{-2x^2}{x^4 + 1}\), the numerator is of degree 2, while the denominator is degree 4.
When the numerator's degree is smaller, the horizontal asymptote tends towards \(y = 0\), as the value of the function diminishes towards zero with increasing \(|x|\). This comparison is essential for predicting how the function behaves when extending out towards positive or negative infinity.

• Degree of numerator: 2
• Degree of denominator: 4
• Results in a horizontal asymptote at \(y = 0\)