Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$\begin{array}{l}y_{1}=\ln \left[x^{2}(x-4)\right] \\\y_{2}=2 \ln x+\ln (x-4)\end{array}$$

Step-by-Step Solution

Answer

Upon graphing and creating tables of values for the functions \(y_{1}=\ln[x^{2}(x-4)]\) and \(y_{2}=2 \ln x+\ln (x-4)\), the graphical and tabular results suggest that the two functions are equal which is then verified algebraically.

Step 1: Graph the functions

Using a graphing utility, input the two functions \(y_{1}=\ln\left[x^{2}(x-4)\right]\) and \(y_{2}=2 \ln x+\ln (x-4)\) to produce the graphs. The x-values should be in the domain of the functions, where \(x>4\) for \(y_{1}\) and both \(x>0\) and \(x≠4\) for \(y_{2}\).

Step 2: Create a table of values

Next, use the table feature of the graphing utility to produce a table of values for each function. Be sure to use the same x-values as used for graphing to ensure that the plotted points correspond to the graph of the function.

Step 3: Analyze graphs and tables

Examine both the graphs and the table of values. Notice if there are similarities or differences between the two functions. It may be observed that both graphs intersect at certain points which can also be verified by noticing that the two functions have the same y-values at certain x-values in the table.

Step 4: Verify Algebraically

To confirm the observations from the graphs and table, algebraically verify the equality of the two functions by using the properties of logarithms. Starting with the second function, \(y_{2}=2 \ln x+\ln (x-4)\), use the property of logarithms that states the sum of two logarithms is the logarithm of the product of their numbers (i.e, \(\ln a + \ln b = \ln (ab)\)) to rewrite it as \(y_{2}=\ln [x^2(x-4)]\), which is identical to the first function \(y_{1}\). So the two functions are indeed equal.

Key Concepts

Algebraic Verification

Algebraic verification involves proving a hypothesis or conclusion by using algebraic properties and techniques rather than relying solely on visual or numerical data. In this exercise, we checked the equality of two logarithmic functions. Both given as:

• \(y_1=\ln[x^2(x-4)]\)
• \(y_2=2\ln x+\ln(x-4)\)

To verify their equality, the properties of logarithms were utilized. The property that allows the transformation of the sum of logarithms into the logarithm of a product is particularly useful here. So we changed \(y_2\) from a sum into a product inside a single logarithm: \(y_2=\ln [x^2(x-4)]\). This showed that \(y_2\) simplifies to \(y_1\), proving both functions are identical. Such algebraic reasoning confirms graphical or numerical conclusions by providing a solid foundational check.

Domain of Functions

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Logarithmic functions, like the ones in this example, have specific domains dictated by their inherent properties. A function \(\ln x\) is only defined for \(x > 0\) because the logarithm of zero or negative numbers is undefined.For our exercise, the domains are crucial:

• In \(y_1=\ln[x^2(x-4)]\), the function is defined when \(x^2(x-4)>0\), which simplifies to \(x>4\).
• For \(y_2=2\ln x+\ln(x-4)\), both parts \(\ln x\) and \(\ln(x-4)\) need to be defined individually which means \(x>4\) but specifically doesn't equal 4 for either part to be defined (as \(\ln 0\) is not possible).

Understanding the domain helps in correctly graphing and analyzing the function's behavior over its applicable range.

Graphing Utility

Graphing utilities, often in the form of calculators or specialized software, are powerful tools for visualizing mathematical functions. When graphing logarithmic functions such as \(y_1=\ln[x^2(x-4)]\) and \(y_2=2\ln x+\ln(x-4)\), inputting them into a graphing utility allows you to view the graphs of these equations over a specified range of x-values. To effectively use a graphing utility:

• Input each function separately to see individual graphs.
• Adjust the graphing window to include the domain of the functions, focusing mainly on x-values greater than 4 where both functions are valid.
• Use features such as zoom and pan to examine key features of the graphs closely, such as intersections or asymptotic behavior.

Graphing utilities support visually confirming findings and provide immediate visual feedback on function behavior.

Table of Values

Creating a table of values is a method to numerically explore a function and verify points on its graph. When a graphing utility generates a table of values for a given function, it presents input x-values alongside their corresponding y-values.To effectively use a table of values for functions \(y_1\) and \(y_2\):

• Select x-values mainly from the domain (>4) to ensure valid computation.
• Compare y-values of both functions at identical x-values to detect any equalities or patterns.
• Look for specific x-values where the y-values coincide, suggesting points of intersection.

Table of values serves as a check for graphical methods and plays a crucial role in identifying characteristics of a function numerically.