Problem 27 - College Algebra

Question

Find the vertical asymptotes, if any, of the graph of each rational function. $$r(x)=\frac{x}{x^{2}+4}$$

Step-by-Step Solution

Answer

The given rational function has no vertical asymptotes.

Step 1: Identify the denominator of the rational function

In this rational function, \(x^{2}+4\) is the denominator.

Step 2: Set the denominator equal to zero and solve for x

The vertical asymptotes occur where the denominator of the rational function equals zero, as the function becomes undefined at these points. So, we set \(x^{2}+4 = 0\), but there are no real roots for this equation since \(x^{2}= -4\) would require taking the square root of a negative number.

Step 3: Conclusion for the vertical asymptotes

Since there are no real roots for the equation, we conclude that no vertical asymptotes exist for the given rational function.

Key Concepts

Rational Functions

A rational function is a type of function that is formed by dividing two polynomial functions. In simpler terms, it is a fraction where the numerator and the denominator are both polynomials.

• The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
• Rational functions are important in mathematics because they can model various real-world situations and provide insights into the behavior of complex systems.

Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and discontinuities. Vertical asymptotes occur in particular where the denominator is zero and the numerator is not zero.

Denominator

In any rational function, the denominator plays a crucial role in determining where the function may not be defined. Specifically, a rational function is undefined wherever its denominator equals zero.

• In the expression \( r(x) = \frac{x}{x^2 + 4} \), the denominator is \( x^2 + 4 \).
• It's essential to identify these points because they can tell us where a potential vertical asymptote might appear if the function becomes undefined.

Knowing that the denominator cannot be zero provides a method for locating vertical asymptotes by solving the equation obtained from setting the denominator to zero.

Solving Equations

Solving equations is a fundamental skill in finding the vertical asymptotes of rational functions. The primary goal is to set the denominator equal to zero and solve for the variable. This process determines where the function is undefined.

• For the function \( r(x) = \frac{x}{x^2 + 4} \), you solve \( x^2 + 4 = 0 \) to find potential vertical asymptotes.
• Attempting to solve this equation reveals that there are no real solutions, as the square root of a negative number is not possible in the set of real numbers.

This outcome leads to the conclusion that there are no vertical asymptotes, and the function is defined for all real \( x \).

Real Roots

Real roots are solutions to equations that locate points where the function may be undefined or change behavior dramatically, such as at vertical asymptotes. Real roots are values of \( x \) that satisfy the equation when the denominator is set to zero.

• Identifying real roots involves solving the equation obtained from setting the denominator to zero.
• In the exercise, the equation \( x^2 + 4 = 0 \) has no real solutions since solving leads to \( x^2 = -4 \), and square roots of negative numbers are not real.

Therefore, the lack of real roots in this situation confirms that there are no vertical asymptotes for the function \( r(x) \). Real roots provide crucial insight into the places where the rational function would be undefined.