Question

If the graph of a rational function \(R\) has the vertical asymptote \(x=4,\) the factor \(x-4\) must be present in the denominator of \(R .\) Explain why.

Short answer

The factor \(x - 4\) in the denominator is needed to create a vertical asymptote at x=4 because it forces the denominator to zero at that point.

Step-by-step solution

- Understanding Asymptotes

A vertical asymptote in the graph of a rational function indicates that at that specific value of x, the function tends to infinity. This happens when the denominator of the function approaches zero at that x-value because division by zero is undefined.

- Expression for Vertical Asymptote

If the graph has a vertical asymptote at x=4, it means that when x approaches 4, the function's denominator must approach zero. So, the denominator must have a factor that equals zero when x=4.

- Identifying the Factor

The factor in the denominator that equals zero when x is 4 is \(x - 4\). Including this factor ensures the function has a vertical asymptote at x=4, as required. Hence, \(x - 4\) must be a factor in the denominator of the rational function R.

Key concepts

Rational Function

A rational function is a type of mathematical function represented by the ratio of two polynomials. It is written in the form, \[ R(x) = \frac{P(x)}{Q(x)} \], where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) ≠ 0. Rational functions can have interesting properties such as vertical asymptotes, horizontal asymptotes, and holes. These occur where the function is undefined due to issues like division by zero. To better understand the unique behavior of rational functions, we need to explore their denominators and what happens when these denominators approach zero.

The key aspect of rational functions is the relationship between the numerator and the denominator. Changes in the denominator, especially when it equals zero, can lead to vertical asymptotes, where the function values will shoot off towards positive or negative infinity.

Denominator Factor

In a rational function, the denominator plays a crucial role in determining the behavior of the function, especially when it approaches zero. For example, consider the rational function \( R(x) = \frac{P(x)}{(x-4)} \). Here, the denominator is \( x - 4 \), which becomes significant.

The factor \( x - 4 \) in the denominator directly affects the vertical asymptote of the function. When \( x = 4 \), the denominator becomes zero, \( Q(x) \rightarrow 0 \), causing the function to be undefined. This undefined behavior is what creates a vertical asymptote. Consequently, to identify vertical asymptotes, one should look for factors in the denominator that can become zero. These factors are visible till the point where the entire polynomial equals zero. For example, if a rational function has a vertical asymptote at \( x = 4 \), the denominator must include a factor of \( x - 4 \). This explains why the function tends towards infinity at \( x = 4 \)—because the division by zero is undefined.

Undefined Division

Division by zero is a critical concept in mathematics that frequently turns up in rational functions. When the denominator of a function equals zero, it leads to an undefined expression. For instance, in the rational function \( R(x) = \frac{P(x)}{(x-4)} \), when \( x = 4 \), we get \( R(4) = \frac{P(4)}{0} \), which is undefined.

This undefined division results in vertical asymptotes, as the function cannot produce a valid numerical output at that specific value of \( x \). Therefore, to find where a rational function becomes undefined, set the denominator equal to zero and solve for \( x \). These solutions are the x-values where the function has vertical asymptotes. For example, if you know \( x = 4 \) is where the function is undefined, then it is clear that \( x - 4 \) must be a factor in the denominator. This ensures the division is undefined at \( x = 4 \), thus creating a vertical asymptote.

• Remember, vertical asymptotes occur with undefined divisions.
• Solution to denominator equals zero gives the location of asymptotes.
• Vertical asymptotes lead to rapid increase or decrease of function values near the undefined point.