Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\ln \frac{x+2}{x-1}$$

Step-by-Step Solution

Answer

Note: Actual results may vary depending on the type of graphing utility used. The graphical solution will give an approximate domain and maximum or minimum values, whereas an analytical solution can provide exact results.

Step 1: Graphing the function

The first step will be to use a graphing utility to plot the function \(\ln \frac{x+2}{x-1}\). The graph will help visualize the direction of the function and point out the intervals where the function increases or decreases.

Step 2: Finding the domain

The function is undefined at \(x=1\) because the denominator of the function is 0 at \(x=1\). Also, the argument of the logarithmic function must be positive, i.e., \(\frac{x+2}{x-1} > 0\). Solving this inequality will give the domain of the function.

Step 3: Identifying increasing or decreasing intervals

By closely looking at the graph of the function, we can identify the intervals. The function is increasing when the slope of the tangent line at any point on the function’s curve points upwards, and decreasing when the slope of the tangent line points downwards.

Step 4: Approximating relative maximum or minimum

A relative maximum is the highest point in a particular section of the graph, whereas a relative minimum is the lowest point. These can be found by looking for peaks and valleys in the function graph. Use the graph plotted in Step 1 to identify these points and approximate their values.

Key Concepts

Domain of a Function

The domain of a function is crucial for understanding where a function is defined and can produce a real number output. For the function \(f(x) = \ln \frac{x+2}{x-1}\), it is important to ensure that the expression inside the logarithm, \(\frac{x+2}{x-1}\), is greater than zero, as logarithms of non-positive numbers are undefined.
To find this domain, solve the inequality \(\frac{x+2}{x-1}>0\). This means:

• \(x+2>0\) and \(x-1>0\), which implies \(x>-2\) and \(x>1\)
• or \(x+2<0\) and \(x-1<0\), which is not possible since there are no values of \(x\) that satisfy both conditions simultaneously.

Continuing with \(x>1\) ensures that the fraction is positive. Therefore, the domain of the function is the interval \((1, \infty)\). This means the function is undefined and cannot have real values at \(x = 1\).

Increasing and Decreasing Functions

To determine where the function is increasing or decreasing, inspect the graph plotted from the previous step. The function is increasing in an interval where the slope of the tangent to the curve is positive. Conversely, it is decreasing where the slope is negative.
In practice, for the function \(f(x)=\ln \frac{x+2}{x-1}\):

• Evaluate the derivative, \(f'(x)\), to find where it is positive or negative.
• Examine the sign of \(f'(x)\) on intervals divided by critical points, found by solving \(f'(x)=0\) or where \(f'(x)\) is undefined.

Based on the graph, you can visually divide the domain and ascertain these intervals. The direction of the curve will indicate whether it's rising (increasing) or falling (decreasing). For this specific function, ensure you note these intervals to know the behavior across its domain.

Relative Extrema

Relative extrema refer to the points on a graph where the function reaches a local maximum or minimum compared to neighboring points.
For \(f(x)=\ln \frac{x+2}{x-1}\), look for points where the derivative \(f'(x)\) changes sign, indicating potential relative extrema. At these points:

• A change from positive to negative identifies a relative maximum.
• A change from negative to positive identifies a relative minimum.

Relative extrema can be approximated using the graph created earlier. Their exact locations can typically be computed analytically, but rounding off values, as per the original exercise, to three decimal places might be necessary for detailed answers.

Logarithmic Functions

Logarithmic functions are based on logarithms, specifically the natural logarithm \(\ln(x)\), which uses the base \(e\), a fundamental irrational constant approximately equal to \(2.718\). These functions have particular characteristics and behaviors.
When working with the function \(f(x)=\ln \frac{x+2}{x-1}\), this involves understanding how the function’s argument, \(\frac{x+2}{x-1}\), determines its properties:

• Logarithms are only defined for positive arguments hence, \(\frac{x+2}{x-1}>0\). This affects the domain.
• The natural logarithm has a vertical asymptote, marking where the function approaches infinity or negative infinity.

Understanding these properties is key for graphing, which reveals the important features and intervals of behavior for logarithmic functions like \(f(x)\). Awareness of these features assists in analyzing how the function increases or decreases and the presence of any relative extrema.