Problem 37 - Precalculus with Limits

Question

In Exercises 33-38, (a) use a graphing utility to graph the function and find the zeros of the function and (b) verify your results from part (a) algebraically. \(f(x) = \frac{3x-1}{x-6}\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{3x-1}{x-6}\) has one zero which is \(x = \frac{1}{3}\).

Step 1: Graph the Function and Determine the Zeros

Using a graphing utility, input the function \(f(x) = \frac{3x-1}{x-6}\). From the graph, locate the x-values where the function touches the x-axis. These are the zeros of the function, values of x for which \(f(x)=0\). Remember that there is a vertical asymptote at x=6, because the denominator of the function becomes zero at this value.

Step 2: Algebraically Determine the Zeros

Set the function equal to zero and solve for x to verify the zeros found in step 1. \[0= \frac{3x-1}{x-6}\]To solve for x, the equation can be multiplied on both sides by (x-6) to get rid of the denominator on the right:\[0 * (x-6) = (3x-1)\]This simplifies to:\[0 = 3x - 1\]Solving for x gives the zero:\[x = \frac{1}{3}\]This is our only zero, as the graph has only a single crossing point with the x-axis. Note that the value x=6 is not a zero of the function, but a place where the function is undefined due to the denominator becoming zero.

Key Concepts

Graphing Rational Functions

Graphing a rational function such as \(f(x) = \frac{3x-1}{x-6}\) is an essential skill in understanding the behavior of the function across the coordinate plane.
To begin graphing, you can use a graphing utility to input the function. This tool will graph the function by plotting various points on the plane, providing you with a visual representation. A typical rational function graph may exhibit different characteristics, such as curves, asymptotes, and intercepts.

• When graphing, notice important features like where the graph intersects the axes.
• Identify any asymptotes, which are lines that the function approaches but never touches.
• Look for the intercepts — points where the graph crosses the axes.

These features provide crucial information about the function, such as its domain, range, and potential discontinuities. For functions with rational expressions, understanding the graph can help visualize zeros and undefined points, thereby indicating where further algebraic verification is necessary.

Finding Zeros of Functions

Finding zeros of a function refers to identifying the x-values at which the function equals zero. For the function \(f(x) = \frac{3x-1}{x-6}\), finding zeros means solving for \(x\) where \(f(x)=0\).
To find the zeros algebraically, set the function equal to zero and solve for \(x\): \[ 0 = \frac{3x-1}{x-6} \]
From here, eliminate the fraction by multiplying both sides by \((x-6)\), giving \[ 0 = 3x - 1 \]
Solving this equation results in a zero at \(x = \frac{1}{3}\). This means at \(x = \frac{1}{3}\), the graph of the function will intersect the x-axis, indicating the point where the value of the function becomes zero.

Finding zeros is crucial in graphing and understanding function behavior as it underscores the roots and x-intercepts of the function. These roots are vital in solving real-world problems where functions model practical phenomena.

Vertical Asymptotes

A vertical asymptote of a rational function occurs where the function is undefined. Typically, this is where the denominator equals zero, as the division by zero is undefined. For \(f(x) = \frac{3x-1}{x-6}\), the vertical asymptote is found by identifying where the denominator \(x-6\) equals zero.
Set the denominator equal to zero and solve for \(x\): \[ x - 6 = 0 \]
This calculation results in \(x = 6\).

Vertical asymptotes are represented on the graph as vertical lines that the function approaches but never crosses. In the graph of a rational function like \(f(x)\), they divide the graph into different sections and indicate important changes in the function's behavior. As \(x\) approaches the value of the asymptote, the function values tend to increase or decrease without bound, leading to a sharp division on the graph.

• Vertical asymptotes help in understanding the limits and undefined areas of the function.
• They play a crucial role in distinguishing real-world boundaries when interpreting mathematical models.

These concepts help us capture the dynamic behavior of rational functions in various contexts.