Problem 70 - Precalculus with Limits

Question

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window. \(f(x)=\ln (x+2)\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\ln(x+2)\) starts at \(x=-2\) and goes off to the right forever, appearing as a soft curve that extends upward and the right.

Step 1: Identify the Domain

Understand that the function \(f(x)=\ln(x+2)\) has a domain of \(x>-2\). This because logarithmic functions can only take positive arguments; if \(x+2\) was less than or equal to zero, then it would not be valid for the function. So, for \(f(x)=\ln(x+2)\), \(x\) must be greater than -2. This means \(x\) can be any value in the interval \(-2 < x < ∞\).

Step 2: Set the Window Parameters

Given the domain of \(x \in (-2, ∞)\), use the graphing utility to set the window parameters. As a starting point, try setting the minimum \(x\) value to, for example, -2 and the maximum to, let’s say, 10. The range (y-values) can be subjective; you might try from -5 to 5 for starters. Adjust if necessary based on how the graph looks.

Step 3: Graph the Function

Now, input the function \(f(x)=\ln(x+2)\) into the graphing utility and plot the function within the specified window. Ensure you're able to see the nature of the function clearly. You should see a graph that is undefined for \(x \leq -2\), and the graph should look like a soft curve that extends upward and the right.

Key Concepts

Logarithmic Function Domain

When we talk about the domain of a logarithmic function, such as f(x) = \(\ln(x+2)\), we're referring to the set of all possible input values (or x-values) that the function can accept. A fundamental property of logarithms is that they are only defined for positive arguments. As a result, within the expression \ln(x+2), the value of (x+2) must be strictly greater than zero – meaning that x itself must be greater than -2. Consequently, the domain for f(x) is x > -2, extending all the way up to positive infinity, which can be expressed as (-2, \infty) in interval notation.

In practical terms, this means that if you attempt to plug a value of x that is less than or equal to -2 into the function, the output will be undefined because you cannot take the logarithm of a non-positive number. Understanding the domain is not just a mathematical formality; it guides us in choosing appropriate values for graphing and ensures the function's behavior is accurately represented.

Graphing Utility Use

Graphing utilities, such as graphing calculators or computer software, are powerful tools that allow us to visualize mathematical functions, including logarithmic ones. To effectively use a graphing utility, it's important to input the function correctly and understand the features that the utility offers. For the given function f(x) = \(\ln(x+2)\), we would enter it just as it appears, ensuring to include the parentheses to maintain order of operations. After entering the function, you can utilize zoom features, trace functions to analyze specific points on the graph, and even evaluate the function at particular x values. A graphing utility can provide an immediate visual representation, which is immensely helpful for grasping the growth and behavior of the function – especially where it starts to curve and approach its asymptotes. By making good use of a graphing utility, you get to transform abstract logarithmic equations into tangible curves that are easier to analyze and understand.

Viewing Window Settings

Setting the appropriate viewing window is a crucial step in graphing functions. The viewing window refers to the range of x and y values that are visible on your graph. For the function f(x) = \(\ln(x+2)\), we need to consider its domain which starts at x = -2. Therefore, our window should start slightly to the left of -2 (to make the vertical asymptote visible) and extend far enough to the right to observe the behavior of the curve, possibly set to x = 10 or more if needed. Vertically, the function can take on both positive and negative values, so we should include both in our viewing window.

Starting with a symmetrical window such as y = -5 to y = 5 is useful because it centers the graph around the horizontal axis, but adjustments are often necessary. If the function appears flattened or too steep, tweaking the window settings is required to capture the true nature of the curve. Incremental adjustments to both the x- and y-axes can yield a graph that is detailed and sufficiently scaled to illustrate the function's key characteristics. A well-chosen window can make the difference between a graph that clarifies a function’s properties and one that obscures them, so taking the time to fine-tune your window settings is an important part of graphing.