Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

When the graph of a rational function \(f\) has a vertical asymptote at \(x=4,\) can \(f\) have a common factor of \((x-4)\) in the numerator and denominator? Explain.

Step-by-Step Solution

Answer

No, the function \(f\) cannot have a common factor of \((x-4)\) in both the numerator and the denominator. If \((x-4)\) were a common factor, it would cancel out and \(x=4\) would not be a place where the function is undefined, therefore it will not be a vertical asymptote.

Step 1: Understanding Rational Functions and Asymptotes

A rational function is of the form \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomial functions. A vertical asymptote occurs where the denominator, \(q(x)\), is zero, and the numerator, \(p(x)\), is nonzero.

Step 2: Analyze the Function Given

In our case, it's stated that the function \(f\) has a vertical asymptote at \(x=4\). This implies that \(q(x)\) must be zero at \(x=4\) and \(p(x)\) should not be zero at \(x=4\). Therefore, the factor \(x-4\) can be found as a factor in the denominator, but it cannot be a common factor in both the numerator and denominator.

Step 3: Conclusion

If \(x-4\) is a common factor in both the numerator and the denominator, it would cancel out. Therefore, \(x=4\) would not be a place where the function is undefined, it will not be a vertical asymptote. Therefore, \(f\) cannot have a common factor of \((x-4)\) in the numerator and denominator.

Key Concepts

Rational Function

Understanding rational functions is key to mastering algebra and calculus. A rational function is a type of function that can be expressed as the ratio of two polynomial functions. In simpler terms, it's like a fraction where both the numerator and the denominator are polynomials. The general form of a rational function is
\begin{align*}f(x) = \frac{p(x)}{q(x)}\end{align*}where \(p(x)\) and \(q(x)\) must be polynomials. Importantly, the denominator, \(q(x)\), cannot be zero, as division by zero is undefined.

Understanding this definition helps you observe key features in a rational function, including its domain, range, and potential asymptotes. Always remember that the behavior around the values that make the denominator zero is especially important, leading us into the concept of asymptotes.

Asymptote Definition

The term asymptote might sound daunting, but it’s an essential concept in understanding the behavior of graphs in algebra. An asymptote is a line that a graph approaches but never actually reaches. It acts like a boundary for the graph of a function. There are different types of asymptotes, such as horizontal, vertical, and oblique.

A vertical asymptote, which is what we’re focusing on here, is a vertical line that corresponds to the x-values that make the denominator of a rational function zero (assuming the numerator does not also become zero at the same point). The equation of a vertical asymptote is always \(x = a\), where \(a\) would cause the denominator to be zero. As x-values approach \(a\) from either side, the values of the function either rise or fall without bound, indicating the presence of a vertical asymptote.

Polynomial Functions

If rational functions are fraction-like, then polynomial functions are its building blocks. A polynomial function is composed of constants and variables that are only combined using addition, subtraction, multiplication, and non-negative integer exponents. It’s written in the form:
\begin{align*}p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\end{align*}where each \(a_i\) represents a coefficient and \(n\) is the degree of the polynomial, indicated by the highest exponent of \(x\).

Polynomials are generally simpler to handle than rational functions because they don't involve division by variable expressions, meaning they don’t have asymptotes. However, their behavior is still rich and interesting, with features like intercepts, turning points, and end behavior.

Graph of Rational Function

The graph of a rational function is a visual representation that shows what the function looks like on a coordinate plane. It can be quite complex due to the properties of the rational function. When graphing, we look for intercepts, horizontal and vertical asymptotes, holes (points where there’s a common factor in the numerator and denominator), and regions where the function is increasing or decreasing.

The exercise we're discussing highlights the importance of looking for vertical asymptotes on such a graph. They are found where the denominator of the function is zero but not the numerator. To graph a rational function properly, you need to identify these critical features and understand how they influence the shape and behavior of the graph around those points. Doing so will provide insight into the function's overall character and potentially help in finding solutions to complex problems.