Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ f(x)=\frac{-4 x}{x^{2}+4} $$

Short answer

The function \(f(x)=\frac{-4x}{x^{2}+4}\) has no vertical asymptotes.

Step-by-step solution

Find the denominator of the function

The denominator of the given function is \(x^{2}+4\). We must analyze this part of the function to see if it ever takes on the value 0 under real numbers.

Set denominator equal to zero

Setting the denominator equal to zero gives us \(x^{2}+4=0\).

Attempt to solve for x

Attempting to solve for x by subtracting 4 from both sides gives \(x^{2}=-4\). Since the square of a real number cannot be negative, there is no real value for x that would make the denominator equal to zero.

Interpret the result

Since there is no real value for x that would make the denominator equal to zero, the function has no vertical asymptotes.

Key concepts

Asymptote Analysis

Understanding the concept of asymptotes is essential for students analyzing the behavior of functions, particularly in calculus and pre-calculus. An asymptote is a line that the graph of a function approaches but never actually touches. Vertical asymptotes occur at values of x where the function is undefined, often corresponding to where the denominator in a rational function is zero.
To determine the vertical asymptotes of a function, we first locate values of x that cause the function's denominator to become zero. In our textbook example, the denominator is the expression \(x^2 + 4\). It seems we might think it could be zero, but upon closer examination, due to the properties of real numbers, we find it is impossible for this expression to be zero. One cannot find a real number x that satisfies the equation \(x^2 = -4\) because the square of any real number is always non-negative. This conclusion tells us that there are no values of x at which the function is undefined, thus resulting in no vertical asymptotes. This analytical process can be summarized as follows:

• Identify the denominator of the function.
• Set the denominator equal to zero to find critical values.
• Test the critical values against the properties of real numbers.
• Determine if the function has any vertical asymptotes based on step 3's outcome.

Students learning this concept for the first time should methodically follow this step-by-step procedure while remembering that vertical asymptotes represent the limitations of what the function can reach given the realm of real numbers.

Rational Function Behavior

Rational functions are expressed as the ratio of two polynomials and their characteristics can greatly vary based on the coefficients and degrees of these polynomials. When investigating the behavior of a rational function, there are several aspects to consider:

• The degree of the numerator and the denominator: The relationship between their degrees can inform us about the end behavior of the function.
• Zeros and poles: Where the function reaches a zero versus where it is undefined (in the form of a pole or asymptote).
• Intervals of increase and decrease: These are determined by analyzing the first derivative of the function.
• Concavity and inflection points: Examined by using the second derivative.

Using the exercise from the textbook as an example, \(f(x) = \frac{-4x}{x^2+4}\), we can observe that the degree of the numerator is less than that of the denominator. This suggests that as \(x\) grows larger in magnitude, the function will approach zero, which is consistent with the horizontal asymptote at \(y=0\). Moreover, since the denominator cannot be zero, there are no vertical asymptotes or poles, and the graph of this function will be smooth and continuous over all real x-values. Understanding these principles enables students to sketch the graph of the function and anticipate its behavior across different domains.

Real Number Properties

Real numbers are the foundation of algebra and calculus, encompassing counting numbers, zero, fractions, decimals, and irrational numbers. Key properties of real numbers include the following:

• Order Property: Any two real numbers can be compared, and one will always be greater, less, or equal to the other.
• Closure Property: The sum or product of any two real numbers is a real number.
• Additive and Multiplicative Identity: Adding zero or multiplying by one leaves any real number unchanged.
• Additive and Multiplicative Inverse Property: Each real number has an opposite and a reciprocal (except zero, which only has an opposite).
• Property of Non-negativity of Squares: The square of any real number is non-negative. Diving into this property reveals why the function in our exercise \(f(x) = \frac{-4x}{x^2+4}\) has no vertical asymptotes. The expression under the square root, \(x^2\), cannot produce a negative number, thereby negating the possibility of zeroing the denominator.

For students to successfully analyze function behaviors and limit behaviors such as asymptotes, they must be proficient in these properties. They serve as the logical underpinnings of why certain solutions to equations are not just improbable, but impossible within the set of real numbers.