Problem 68 - Contemporary Precalculus

Question

Find a viewing window (or windows) that shows a complete graph of the function. $$r(x)=\ln \left(e^{x}\right)$$

Step-by-Step Solution

Answer

Question: Determine a suitable viewing window for the function $$r(x) = \ln \left(e^{x} \right).$$ Answer: A suitable viewing window for the function $$r(x) = \ln \left(e^{x} \right)$$ is the horizontal axis (x) ranging from -10 to 10 and the vertical axis (r(x)) ranging from -10 to 10.

Step 1: Simplify the function, if possible.

In this case, we have the natural logarithm function (denoted as ln) and the exponential function with base e (represented by e^x). We know that taking the natural logarithm of e to the power of x (ln(e^x)) allows us to simplify the given function, using the property $$\ln \left(e^x \right) = x.$$ Therefore, our function can be expressed as: $$r(x) = x.$$

Step 2: Find the domain and range.

Now, let's find the domain and range for the simplified function. For the function $$r(x) = x$$, the domain is all real numbers since we can input any real value of x for the function. Thus, the domain is: $$(-\infty, +\infty).$$ The graph of the function also represents a straight line with a 45-degree angle, so the range of the function (the possible values of r(x)) is also all real numbers: $$(-\infty, +\infty).$$

Step 3: Choose a suitable viewing window.

A standard viewing window in most graphing utilities is (-10, 10) for both the horizontal and vertical axes, however, the given function $$r(x) = x$$ can theoretically extend infinitely in both directions, for x and r(x). One of the most common windows to show a complete graph of $$r(x) = x$$ might be: Horizontal axis (x): $$[-10, 10]$$ Vertical axis (r(x)): $$[-10, 10]$$ This viewing window is suitable for a first approach, but other windows could be used as well. The choice of a specific window may depend on preferences or specific requirements, but the main idea is to capture the behavior of the function and grasp its characteristics.

Key Concepts

Natural Logarithm

Understanding the natural logarithm is crucial when dealing with logarithmic functions. The natural logarithm, often denoted as 'ln', is a special logarithm where the base is the mathematical constant 'e', approximately equal to 2.71828. When we say natural logarithm, we refer to the power to which 'e' must be raised to obtain a certain number.

The natural logarithm has several important properties, and one of them is its inverse relationship with the exponential function. For instance, the expression \( \ln(e^x) \), essentially asks the question: 'To what power must we raise 'e' to get \( e^x \)?' The answer is simply 'x', showing the function and its inverse undo each other. This property is why the function from the exercise simplifies so neatly.

When graphing natural logarithmic functions, it's helpful to remember that they pass through the point (1,0), grow slower as they move to the right, and are undefined for values less than or equal to zero, which gives them a distinct 'L' shape on the graph.

Exponential Function

In parallel to logarithmic functions, exponential functions play a pivotal role in various scientific fields. These functions are of the form \( e^x \), where 'e' represents Euler's number, and 'x' is the exponent. The result of an exponential function is how much we multiply the base 'e' for itself 'x' times. This exponential function demonstrates continual growth (or decay, if the exponent is negative), which is why it's often used to model phenomena like population growth or radioactive decay.

The graph of an exponential function like \( e^x \) is characterized by a rapid increase as 'x' gets larger, a horizontal asymptote (usually the x-axis), and the fact that it's always positive. It's important to note, too, that the exponential function is the inverse of the natural logarithm, which means they are closely interconnected. To graph an exponential function accurately, it is essential to depict its ever-increasing nature.

Domain and Range of Functions

The concepts of domain and range are fundamental in understanding any function. The domain of a function includes all possible input values, while the range refers to all possible outputs.

For the exercise's function \( r(x) = \ln(e^x) \), the domain starts off as all real numbers, because the exponent 'x' in an exponential function can take any real value. After simplification to \( r(x) = x \), the domain remains unchanged. However, when dealing with just the logarithmic part, the domain would be restricted to positive numbers only, since the logarithm of zero or a negative number is not defined.

The range of the function represents the set of all output values. For \( r(x) = \ln(e^x) \), before simplification, the range would have been all real numbers because the logarithm can output any number given a positive input. After simplification to \( r(x) = x \), the range, just like the domain, is all real numbers, evident by its straight-line graph. To choose an effective viewing window for graphing, consider the behavior of the function, and ensure the window includes significant features like intercepts, turns, or asymptotes, to accurately portray the function's character.