Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}}{1-x^{2}}$$

Step-by-Step Solution

Answer

Domain: \( x \neq \pm 1 \); Vertical asymptotes at \( x = \pm 1 \); Horizontal asymptote at \( y = 1 \).

Step 1: Identify the Domain

The function is given as \( f(x) = \frac{x^2}{1-x^2} \). The denominator \( 1-x^2 \) cannot be zero, so we solve \( 1-x^2 = 0 \) which gives \( x = \pm 1 \). Thus, the domain is all real numbers except \( x = 1 \) and \( x = -1 \).

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero. From \( 1-x^2 = 0 \), we previously found \( x = 1 \) and \( x = -1 \). Hence, \( x = 1 \) and \( x = -1 \) are vertical asymptotes.

Step 3: Determine the Horizontal Asymptote

For rational functions, if the degree of the numerator equals the degree in the denominator, the horizontal asymptote is the ratio of the leading coefficients. Here, both numerator and denominator have degree 2, with leading coefficients 1 each. Therefore, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

Step 4: Analyze Critical Points and End Behavior

Since \( f(x) = \frac{x^2}{1-x^2} \) has \( x^2 \) in the numerator, the graph will tend towards \( +\infty \) as \( x \to \pm \infty \). The function is even, symmetric around the y-axis.

Step 5: Plot the Function including Asymptotes

Draw the vertical asymptotes at \( x = 1 \) and \( x = -1 \), and the horizontal asymptote at \( y = 1 \). Since \( x^2 \) is always non-negative, the function stays above the x-axis. The graph approaches the horizontal asymptote from above. The function crosses the y-axis at \( (0,0) \).

Key Concepts

Asymptotes

In rational functions, asymptotes represent important boundaries the graph will approach but not actually reach. For vertical asymptotes, these occur at values of the domain where the denominator equals zero, which makes the function undefined. In the given function, the expression in the denominator, \(1 - x^2\), must never be zero. We solve \(1 - x^2 = 0\) to find the vertical asymptotes, obtaining \(x = 1\) and \(x = -1\). Thus, the graph will have vertical lines at these x-values, showing where the function abruptly changes from positive to negative or vice versa.

Horizontal asymptotes are a bit different. They concern the behavior of a function as \(x\) approaches infinity or negative infinity. When the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is found by dividing the leading coefficients. In our example, both polynomials have degree 2, and their leading coefficients are 1. So, the horizontal asymptote is \(y = 1\). As \(x\) becomes very large or very small, the graph of the function will approach this line.

Graphing Rational Functions

Graphing rational functions involves carefully considering both their end behavior and specific points of interest like asymptotes and intercepts.

To graph the function \(f(x)=\frac{x^2}{1-x^2}\), begin by drawing the vertical asymptotes at \(x = 1\) and \(x = -1\). These lines highlight where the graph will sharply increase or decrease without ever crossing them. Then, mark the horizontal asymptote at \(y = 1\). This line indicates where the graph tends to level out as \(x\) moves towards infinity.

Next, address the intercepts. Set the equation \(f(0) = \frac{0}{1 - 0}\), showing the y-intercept at \((0,0)\). Because the numerator \(x^2\) cannot be negative, the graph will reside above or on the x-axis.

With these elements plotted, the sketch of the graph should reflect a gradual approach to the horizontal asymptote and a drastic movement away from vertical asymptotes.

Function Analysis

Analyzing the function \(f(x)=\frac{x^2}{1-x^2}\) gives insights into its symmetry and behavior across its domain. This function is even, meaning it is symmetric about the y-axis. We observe this because substituting \(-x\) for \(x\) gives the same function, \(f(-x) = f(x)\).

An important aspect is end behavior. As \(x\) approaches infinity or negative infinity, \(f(x)\) draws close to the horizontal asymptote, which was found to be \(y = 1\). This reflects the function’s steadiness at extreme values of \(x\).

Moreover, critical points and changes from increasing to decreasing can be predicted by examining the first derivative. Since the given exercise involves understanding without a calculator, key observations such as intercepts, symmetry, and asymptotic behavior guide our interpretation without delving into derivatives directly.

• At \(x = 0\), the function intersects the y-axis at the origin.
• The graph does not exist at \(x = 1\) and \(x = -1\), reinforcing the role of asymptotes as guidelines rather than crossed boundaries.
• The graph trends toward infinity near the vertical asymptotes, indicating sharp increases or decreases.

Overall, rational functions like this illustrate a variety of behaviors dictated largely by their asymptotes and degrees.