Problem 20 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, analytically show that the functions are inverse functions. Then use a graphing utility to show this graphically. $$ \begin{aligned} &f(x)=e^{x}-1 \\ &g(x)=\ln (x+1) \end{aligned} $$

Step-by-Step Solution

Answer

Analytical and graphical validation shows that \( f(x) = e^{x} - 1 \) and \( g(x) = \ln (x+1) \) are inverse functions.

Step 1: Show \(f(g(x)) = x\)

For \(f(g(x)) = x\), substitute \(g(x)\) into \(f(x)\). So, \(f(g(x)) = e^{g(x)} - 1\). Now, replace \(g(x)\) with \(\ln (x+1)\) to get \(f(g(x)) = e^{\ln (x+1)} - 1\). The expression simplifies to \(x + 1 - 1\), which becomes \(x\), as hoped.

Step 2: Show \(g(f(x)) = x\)

For \(g(f(x)) = x\), substitute \(f(x)\) into \(g(x)\). So, \(g(f(x)) = \ln(f(x) + 1)\). Now, replace \(f(x)\) with \(e^{x} - 1\) to get \(g(f(x)) = \ln ((e^{x} - 1) + 1)\). The expression simplifies to \(\ln e^{x}\), which simplifies further to \(x\), as desired.

Step 3: Graphical verification

Graph both \(f(x) = e^{x} - 1\) and \(g(x) = \ln (x+1)\) in the same coordinate plane. The graph of the function \(f(x)\) is a translated version of \(e^{x}\), with a shift of down 1 unit. The function \(g(x) = \ln (x+1)\) represents a logarithmic function, shifted by 1 unit left. If \(f\) and \(g\) are inverse functions, then these two functions should be mirror images of each other with respect to the line \(y=x\). Observing the graph, we can see that this is indeed the case, confirming that \(f(x)\) and \(g(x)\) are inverse functions graphically as well.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. They have the general form of f(x) = a^x where a is a positive constant known as the base. These functions show rapid growth or decay and are widely used in the contexts of compound interest, population dynamics, and radioactive decay, among others.

For example, the function f(x) = e^x is a common exponential function where the base e is an irrational number approximately equal to 2.71828, known as Euler's number. This particular base is chosen because it has unique properties that simplify the mathematical behavior of natural logarithms.

Here's why exponential functions like e^x are key in calculus and beyond:

• They are their own derivative, thus simplifying differentiation.
• Their inverse functions are logarithms, which have their own set of important applications.
• They describe growth and decay processes accurately.

Understanding exponential functions is crucial for solving problems involving exponential growth or decay.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is Euler's number. The natural logarithm of a number x is defined as the power to which e must be raised to obtain that number. In formal terms, if y = ln(x), then e^y = x.

Natural logarithms have several properties that are essential in both theoretical and applied mathematics:

• They are inverses of exponential functions when the base is e.
• They allow for the solving of exponential equations by transforming multiplicative relationships into additive ones.
• They are integral in the study of calculus, particularly in finding areas under curves and solving differential equations.

For instance, the function g(x) = ln(x+1) from the exercise illustrates a shifted natural logarithm, which can be essential for understanding changes in processes described by exponential functions.

Graphical Verification

Graphical verification is a method used to visually confirm mathematical concepts, such as the relationship between functions and their inverses. By graphing two functions in the same coordinate plane, one can inspect certain characteristics and behaviors - such as symmetry with respect to the line y = x for inverse functions.

When using a graphing utility, or creating a graph by hand:

• Look for the line of symmetry; inverse functions should mirror each other relative to the line y = x.
• Examine the domains and ranges; for a function f(x) and its inverse g(x), the range of f(x) should match the domain of g(x) and vice versa.
• Any point (a, b) on the graph of f(x) should correspond to a point (b, a) on the graph of g(x).

In the exercise provided, plotting f(x) = e^x - 1 alongside its purported inverse g(x) = ln(x+1) and looking for these characteristics helps verify the inverse relationship without delving into algebraic proofs. Graphical verification is not only satisfying but also a powerful tool for understanding and communication in mathematics.