Problem 22 - Precalculus with Limits

Question

In Exercises 19-42, use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. \(f(x) = \frac{5}{6}x - \frac{2}{3}x\)

Step-by-Step Solution

Answer

After simplifying the function and graphing, the result is a horizontal line along y=0. Therefore, the function \(f(x) = \frac{5}{6}x - \frac{2}{3}x\) simplifies to \(f(x) = 0\). The graph of this function is a straightforward horizontal line at y=0.

Step 1: Simplify the function

First, simplify the equation. We have \(f(x) = \frac{5}{6}x - \frac{2}{3}x\). If we simplify this, we get that \(f(x) = 0\), as the terms \(\frac{5}{6}x\) and \(\frac{2}{3}x\) cancel out because they're equal but opposite in sign.

Step 2: Use the graphing utility

Now that we have simplified the function, graph it using the graphing utility of your choice. The equation represents a horizontal line at y=0, so that's what should be depicted on the graph.

Step 3: Choose the appropriate viewing window

As the graph is a horizontal line along y=0, choose a viewing window that clearly shows the x and y-axes. Any standard viewing window such as [-10, 10] x [-10, 10] will clearly show the horizontal line.

Key Concepts

Simplifying Functions

Simplifying functions is an essential first step in graphing. In this exercise, the function given is:\[f(x) = \frac{5}{6}x - \frac{2}{3}x\] To simplify, combine like terms. This involves finding a common denominator and subtracting the terms. Here, both fractions share a common base variable, so simplifying results in: \[f(x) = \left(\frac{5}{6} - \frac{4}{6}\right)x = \frac{1}{6}x\] Finally, subtracting the constants in the coefficients results in zero:\[f(x) = 0\]This process shows that any changes in value or operations cancel. Simplifying helps identify the true behavior of the function—in this case, revealing that the graph is a constant horizontal line.

Graphing Utility

A graphing utility is an indispensable tool for visualizing functions. After simplifying, we found that the function represents a horizontal line at \(y = 0\). Graphing utilities, such as graphing calculators or software like Desmos, plot equations visually. Here’s a simple way to proceed:

• Enter the simplified function \(f(x) = 0\) into the graphing tool.
• Adjust settings to show the function as a complete graph.

Using a graphing utility helps see the function at a glance, confirming the behavior as a flat line with no slope or inclination.

Horizontal Line

A horizontal line is a straight line running parallel to the x-axis. For our function, simplifying it to \(f(x) = 0\) confirms the line is along the entire x-axis. In terms of slope, horizontal lines have a slope of zero. This signifies:

• The line does not rise or fall but remains at a constant y-coordinate, here at \(y = 0\).
• It illustrates constant output despite input changes.

Horizontal lines are often used to represent stable, unchanging values across the domain of a function. This visualization is essential for understanding uniformity in datasets or equations.

Viewing Window

Choosing the correct viewing window ensures that the graph accurately represents the function. For the equation \(f(x) = 0\), where the graph is a horizontal line at \(y = 0\), a common choice is:

• A window like \([-10, 10]\) for both \(x\) and \(y\), providing a balanced view of the graph.
• This range helps to capture the axes clearly as well as the interaction or crossing of x and y-axes.

Selecting a viewing window that shows the essentials of the graph is crucial. It simplifies analysis, ensuring that key features like slopes or intercepts are visible without needing to adjust the settings repeatedly.