Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{x^{2}}{x^{2}-4} $$

Short answer

The vertical asymptotes of the function are at x = 2 and x = -2.

Step-by-step solution

Identify the Rational Function

The function provided is a rational function, given by \(f(x) = \frac{x^{2}}{x^{2}-4}\). It is a ratio of two polynomial functions.

Find the Denominator of the Function

The denominator of the function is \(x^{2} - 4\). Ensure that you set this equal to zero.

Solve for x

To find the x-values where the function is undefined (potential vertical asymptotes), solve the equation \(x^{2} - 4 = 0\). This can be achieved by adding 4 to both sides to isolate \(x^{2}\), and then square rooting both sides to solve for x. This results in x = 2 and x = -2.

Check for Holes

Notice that plugging these x-values into the numerator does not result in zero, which means these are indeed vertical asymptotes, not holes. Therefore, confirm that the vertical asymptotes are indeed at x = 2 and x = -2.

Key concepts

Rational Functions

Rational functions are similar to fractions but instead of numbers, they are expressions involving variables. Precisely, a rational function is a ratio of two polynomials where the denominator is not zero. For instance, the function from our exercise,
\[ f(x) = \frac{x^2}{x^2 - 4} \]
is a rational function. The numerator is a polynomial, \( x^2 \), and the denominator is another polynomial, \( x^2 - 4 \). The behavior of the rational function is heavily dependent on the denominator, as it determines when the function is undefined \(cannot be calculated\) due to division by zero.

• If the denominator equals zero, the function has an undefined value.
• When the denominator and numerator have no common factors, the points where the denominator equals zero are vertical asymptotes.
• If the numerator and denominator share a factor, you may find holes instead of vertical asymptotes at the zero points of the shared factor.

Understanding the relationship between the numerator and the denominator is key to analyzing rational functions.

Undefined Function Values

Undefined function values occur where the function cannot produce a valid result. In the case of rational functions, undefined values arise specifically when the denominator is zero, because division by zero is not permitted in mathematics. For our example function,
\[ f(x) = \frac{x^2}{x^2 - 4} \]
we must find values of \( x \) that cause the denominator \( x^2 - 4 \) to be zero. These values are important as they may lead to vertical asymptotes on the graph of the function. In essence:

• Setting the denominator equal to zero gives us the equation \( x^2 - 4 = 0 \).
• Solving for \( x \) gives the potential locations of vertical asymptotes, here \( x = 2 \) and \( x = -2 \).
• Before finalizing these as asymptotes, a check is necessary to determine if these points are actually asymptotes or if the function has holes at these points due to a common factor with the numerator.

In summary, the undefined values of a rational function tell us where to look for significant features such as vertical asymptotes and holes in the graph.

Solving Polynomial Equations

Solving polynomial equations is a fundamental skill in algebra that enables us to find the roots, or solutions, of the equation. These roots represent the x-values where the polynomial equals zero. In the case of rational functions, solving the polynomial set in the denominator equal to zero helps us locate vertical asymptotes. The polynomial equation from our example is:
\[ x^2 - 4 = 0 \]
To solve this:

• First, manipulate the equation to isolate \( x^2 \) by adding 4 to both sides, giving us \( x^2 = 4 \).
• Next, take the square root of both sides to solve for \( x \), resulting in \( x = ±2 \).

Here, \( x = 2 \) and \( x = -2 \) are the solutions, indicating where the denominator of our rational function is zero. With polynomial equations, the technique used to solve them depends on the polynomial's degree and complexity. Factoring, applying the quadratic formula, and synthetic division are some of the methods used for different types of polynomials. Understanding how to solve polynomial equations is crucial in identifying the characteristics of rational functions.