Problem 26 - College Algebra

Question

Find the vertical asymptotes, if any, of the graph of each rational function. $$h(x)=\frac{x}{x(x-3)}$$

Step-by-Step Solution

Answer

The given function \(h(x)=\frac{x}{x(x-3)}\) has a vertical asymptote at \(x=3\).

Step 1: Simplify the Function

Simplify the given function, \(h(x)=\frac{x}{x(x-3)}\), by cancelling the common term in the numerator and the denominator. This results in \(h(x)=\frac{1}{x-3}\).

Step 2: Identify the Asymptotes

Vertical asymptotes occur at the values of x that make the denominator equal to zero. The denominator of this function is zero when \(x=3\). So, x=3 is a vertical asymptote for this function.

Key Concepts

Rational Functions

Rational functions are mathematical expressions that show the ratio of two polynomials. Specifically, they take the form of \(\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not equal to zero. Understanding rational functions is essential as they describe many real-world phenomena, such as rates of change and systems in equilibrium.

When working with rational functions, identifying features like vertical asymptotes can help us understand the behaviour of the graph, especially as the input values get very large or very small. It's essential to note that vertical asymptotes represent the values of \(x\) for which the function cannot produce a result because the denominator is zero, causing undefined behavior in real numbers.

Graph Analysis

Graph analysis of a rational function involves identifying its critical points, intercepts, asymptotes, and end behavior to sketch a comprehensive picture of the function's behavior. A vertical asymptote, such as the one at \(x=3\) for the given function \(h(x)=\frac{1}{x-3}\), suggests the graph will approach this line without ever crossing it.

Interpreting Asymptotic Behaviour

In graph analysis, the vertical asymptote is where the values of the function rise or fall indefinitely, effectively going off to infinity or negative infinity. However, an asymptote does not always mean the function will touch or cross the line; it merely acts as a boundary influencing the graph's shape as \(x\) approaches the asymptote's value.

Simplifying Expressions

Simplifying expressions is a critical skill when working with rational functions. It involves reducing the expression to its simplest form, often by canceling common factors in the numerator and denominator or by factoring polynomials. Simplification helps to avoid unnecessary complexity in calculations and can reveal characteristics of the function that may not be immediately apparent.

For instance, in the solution to the original problem, simplifying \(h(x)=\frac{x}{x(x-3)}\) to \(h(x)=\frac{1}{x-3}\) makes it much clearer that there is a vertical asymptote at \(x=3\), simplifying subsequent steps of the analysis. When simplifying, be mindful to only cancel out factors that are present in both the numerator and denominator to prevent altering the function's properties.