Problem 20 - Precalculus with Limits

Question

In Exercises 19-28, use a graphing utility to graph the inequality. $$y \geq-2-\ln (x+3)$$

Step-by-Step Solution

Answer

The solution to the inequality \(y \geq-2-\ln(x+3)\) is represented by the shaded area above the curve in the graph. This includes all values of \(y\) that are greater than or equal to the value of \(-2-\ln(x+3)\) for each respective \(x\) in the domain \(x>-3\).

Step 1: Identify the Function

The function to be graphed is \(-2-\ln(x+3)\). This is a transformation of the function \(-\ln(x)\), shifted 2 units downwards and 3 units to the left.

Step 2: Create the Graph

Using a graphing utility, plot the function \(y=-2-\ln(x+3)\). The basic curve of \(-\ln(x)\) is reflected over the x-axis, resulting in a decreasing function. It gets shifted to 3 units on the negative x-axis and 2 units down on the negative y-axis. The vertical asymptote will be at x = -3. Because of the negative in front of the logarithm, the graph will be decreasing for x>-3.

Step 3: Apply the Inequality

For the inequality \(y \geq-2-\ln(x+3)\), this means that \(y\) is greater than or equal to the function. This corresponds to the area above the line you graphed. On the graph, shade this area to represent all possible solutions to the inequality.

Key Concepts

Logarithmic Functions

Logarithmic functions are essential for various mathematical and real-world applications. They are the inverse of exponential functions, which means they help us solve equations where the variable is in the exponent. In general, the logarithmic function is written as \(y = \log_b(x)\), where \(b\) is the base of the logarithm, and \(x\) is the argument. The natural logarithm, which is used frequently, has a base \(e\) (approximately 2.718) and is denoted \(\ln(x)\).

When working with logarithmic functions, it's crucial to understand their basic properties:

• They are undefined for non-positive values of \(x\).
• They pass through the point (1,0) when the base is greater than zero.
• They are increasing functions if the base \(b > 1\) and decreasing if \(0 < b < 1\).

Logarithmic functions are also known for their ability to transform multiplicative relationships into additive ones, which is useful in many scientific calculations. In our exercise, the transformation involved creates a new function by adding and reflecting the basic logarithmic curve.

Vertical Asymptotes

A vertical asymptote is a crucial concept when working with graphs of functions. It represents a line that the graph approaches but never touches. For the logarithmic function \(y = \ln(x)\), the vertical asymptote occurs at \(x = 0\) because the logarithm is undefined for zero and negative numbers. This means the graph will rise or fall steeply as it approaches this value.

In our particular function \(y = -2 - \ln(x+3)\), we have modified the basic logarithmic function by shifting it to the left by 3 units. Consequently, the vertical asymptote moves to \(x = -3\). This indicates that as the value of \(x\) approaches -3, the graph of the function will move toward negative infinity. Understanding the location of the vertical asymptote helps in accurately sketching the behavior of the graph along with determining where the function is defined.

Graph Transformations

Graph transformations allow us to modify the appearance of a basic function by applying transformations such as shifts, reflections, stretches, and compressions. When graphing inequalities like \(y \geq -2 - \ln(x+3)\), understanding these transformations is key to identifying how the graph should look.

In the given function, two main transformations take place:

• The graph of \(-\ln(x)\) is shifted downwards by 2 units, which simply moves the entire graph lower on the y-axis.
• It is also shifted 3 units to the left, which affects the horizontal positioning, including the location of the vertical asymptote.
• Reflecting the graph over the x-axis changes it from increasing to decreasing.

Once these transformations are applied, you plot the function and use the inequality sign to determine which part of the plane to shade. For this exercise, \(y\) values greater than or equal to the function are above the graphed curve, indicating that this region is part of the solution to the inequality.