Problem 26 - Algebra and Trigonometry

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each crational function. $$ h(x)=\frac{x}{x(x-3)} $$

Step-by-Step Solution

Answer

The function \(h(x) = \frac{x}{x(x-3)}\) has a vertical asymptote at \(x = 3\), and it does not have any holes.

Step 1: Simplify the function

The given function \(h(x) = \frac{x}{x(x-3)}\) is first simplified by canceling out the common factor \(x\). So the simplified function is \(h(x) = \frac{1}{(x-3)}\)

Step 2: Find vertical asymptote

Next, find the vertical asymptote by setting the denominator equal to zero. So, solve the equation \(x-3 = 0\). The solution \(x = 3\) represents the vertical asymptote. This means as \(x\) approaches 3, the function becomes undefined.

Step 3: Check for holes

To check for a hole, both the numerator and denominator must be equal to zero at the same value of \(x\). For the function \(h(x) = \frac{1}{(x-3)}\), note that there is no value of \(x\) for which both the numerator and denominator equal zero. Therefore, there are no holes in this function.

Key Concepts

Vertical Asymptotes

When graphing rational functions, a vertical asymptote represents where a function approaches infinity. It's like a boundary that the function can never cross. To find vertical asymptotes, we look for values that make the denominator of our rational function equal zero, as these are the values that the function cannot be defined at.

In the given exercise, the vertical asymptote of the function, identified in step 2, is at the point where the denominator, which is \[x-3\], equals zero. Solving \[x-3=0\] gives us \[x=3\], indicating that there is a vertical asymptote at \[x=3\]. As \(x\) approaches 3, the value of the function \(h(x)\) grows larger and larger without bound, confirming the existence of the asymptote.

Holes in a Graph

Holes occur in the graph of a rational function when the same factor in the numerator and denominator zero out each other; however, this common factor must be canceled out before graphing. A hole implies that the function is not defined at this particular \(x\) value but is still continuous at this point.

In step 3 of the solution, we checked for holes by trying to find a factor that zeroes out in both the numerator and denominator. However, it was seen that there is no such factor in the simplified function \(h(x) = \frac{1}{(x-3)}\). Thus, there are no holes in the graph. It's essential to note that simplifying the expression, as seen in step 1 by canceling out \(x\), removes potential holes, which may have existed had we not simplified the function.

Simplifying Rational Expressions

To simplify rational expressions, we often seek to reduce the function to its lowest terms. This could involve factoring the numerator and denominator and then cancelling any common factors between them. Simplification makes it easier to analyze the function and to identify any asymptotes and holes.

In the original exercise, we started by simplifying the function \(h(x) = \frac{x}{x(x-3)}\) by canceling the common factor of \(x\). After simplification, the function became \(h(x) = \frac{1}{(x-3)}\), which is more manageable both algebraically and graphically. It's important to perform this step before moving on to finding asymptotes or holes as simplification can sometimes eliminate factors that result in holes.

Analyzing Functions

Analyzing functions, especially rational ones, involves a few important steps. We must identify vertical asymptotes and holes, determine horizontal asymptote behaviors, find intercepts, and analyze the function's end behavior. Each component provides a deeper understanding of how a function behaves across different regions of its domain.

The step-by-step solution provided a starting point for analyzing our function by simplifying it, finding its vertical asymptote, and checking for holes. This analysis helps predict the graph's attributes — where it will approach infinity, where it's not defined despite seeming continuous, and how it might behave far from the center. These parts create a comprehensive picture needed for graphing rational functions accurately.