Problem 80 - Precalculus : Building Concepts and Connections

Question

Graph \(f(x)=\ln e^{x}\) and \(g(x)=x\) on the same set of axes. (a) What are the domains of the two functions? (b) For what values of \(x\) do these two functions agree?

Step-by-Step Solution

Answer

The domains of both functions are all real numbers. The functions \(f(x)=\ln e^{x}\) and \(g(x)=x\) agree for all values of \(x\), resulting in an infinite number of intersection points, all along the line \(y=x\).

Step 1: Plotting the functions

The function \(f(x)=\ln e^{x}\) is actually equivalent to \(x\) due to the logarithm rules, as taking log base e of e to the power of x gives x. So, both \(f(x)\) and \(g(x)\) are the same lines, graphing them would mean plotting a straight line \(y=x\).

Step 2: Determine the domains of the functions

The domains of the functions \(f(x)\) and \(g(x)\) are all real numbers. For \(f(x)=\ln e^{x}\), the inner function is \(e^{x}\), which accepts all real values for \(x\). The natural logarithm function \(\ln y\) is defined for all positive y, and since \(e^{x}\) is always positive, \(\ln e^{x}\) is defined for all x. For the function \(g(x)=x\), it is a linear function and is defined for all real numbers.

Step 3: Determine where the functions agree

In this case, \(f(x)=\ln e^{x}=x\) and \(g(x)=x\) are the same functions, so they agree on all real numbers, which means they have an infinite number of intersection points, which are all along the line \(y=x\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. A common example is the function \(e^x\), where \(e\) is Euler's number, approximately equal to 2.71828. This type of function plays a crucial role in various fields such as finance, science, and engineering. One key property of exponential functions is their rapid growth or decay, depending on the exponent's sign.

• When the exponent is positive, the function grows quickly as \(x\) increases.
• If the exponent is negative, the function decreases towards zero as \(x\) increases.

In our exercise, \(f(x) = \ln e^x\) breaks down using log rules to simply \(x\), because \(\ln e^x = x\). Here, the exponential function \(e^x\) operates as an intermediary step, reaffirming the importance of understanding these foundational rules.
Understanding exponential functions is pivotal to grasping logarithmic functions, as the latter are the inverse of the exponential ones, thereby solving many real-world problems with logarithms.

Domain of a Function

The domain of a function refers to all the possible input values (or \(x\) values) for which the function is well-defined. Put simply, it tells us what values we are allowed to plug into the function without breaking mathematical rules.

• For the function \(f(x)=\ln e^{x}\), the domain is all real numbers because \(e^{x}\) is defined for every real \(x\), and \(\ln y\) allows for all positive \(y\). Since \(e^{x}\) is always positive, \(\ln e^{x}\) poses no restrictions.
• For the linear function \(g(x) = x\), there are no restrictions on \(x\); thus, its domain is also all real numbers.

Comprehending the domain is critical to understanding which inputs will yield valid outputs. By correctly identifying domains, we ensure that the function behaves as planned over all input values a user might choose.

Linear Equations

Linear equations are the simplest type of algebraic equation. They involve variables raised only to the first power and graph as straight lines with a constant slope. A general form of a linear equation is \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.

• The linear function \(g(x) = x\) is a special case where the slope is \(1\) and the y-intercept is \(0\), resulting in the line passing through the origin with a slope of 45 degrees.

Linear equations serve as an essential building block in mathematics because they represent real-world relationships of proportionality. They simplify evaluation of variables and provide a straightforward visualization through their graph.
In the case of both the functions \(f(x) = \ln e^x\) and \(g(x)=x\), they represent a linear equation, specifically the line \(y=x\), highlighting identical behaviors over all real numbers. Hence, understanding the properties of linear equations allows us to tackle more intricate mathematical challenges effectively.