Question

(a) find the inverse of the function, (b) use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window, and (c) verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\). $$ f(x)=2 \ln (x-1) $$

Short answer

The inverse of the function \(f(x)=2 \ln (x-1)\) is \(f^{-1}(x)=e^{(x/2)}+1\). Both functions when depicted graphically will be mirror images of each other along the line \(y=x\). This is confirmed by verifying that \(f^{-1}(f(x))=x\) and \(f(f^{-1}(x))=x\).

Step-by-step solution

Find the Inverse of the Function

To find the inverse of the function \(f(x)=2 \ln (x-1)\), replace \(f(x)\) with \(y\). This becomes the equation \(y=2 \ln (x-1)\). Swap \(y\) and \(x\) to get the equation \(x=2 \ln (y-1)\). Finally, rearrange the equation to solve for \(y\), which is the inverse function: \[ y = e^{(x/2)} + 1 = f^{-1}(x)\].

Graph Both Functions

Use a graphing utility to graph both functions \(f(x)=2 \ln (x-1)\) and \(f^{-1}(x)=e^{(x/2)} + 1\). These graphs will be mirror images of each other along the line \(y=x\), because they are inverses.

Verification

After obtaining the inverse function and the original function, it's necessary to confirm that \(f^{-1}(f(x))=x\) and \(f(f^{-1}(x))=x\). For \(f^{-1}(f(x))=x\), the inverse \(f^{-1}\) operates on \(f(x)\) to get \(e^{(f(x)/2)} + 1 = e^{((2 \ln (x-1))/2)} + 1 = x\). For \(f(f^{-1}(x))=x\), the function \(f(x)\) operates on \(f^{-1}(x)\) to get \(2 \ln(f^{-1}(x)-1) = 2 \ln(e^{(x/2)} -1) = x\). These confirmations show that the inverse and the original function are equivalent when operated on each other, this means that \(f^{-1}(x)\) is the correct inverse of \(f(x)\).

Key concepts

Finding Inverse Functions

The process of finding an inverse function begins with understanding what an inverse is: essentially, it's a function that reverses another function. To find the inverse, you usually start with a function, say, f(x), and switch the roles of the x and y variables. This means you replace f(x) with y, then solve for y in terms of x. This 'flipped' version is your inverse, which we denote as f^{-1}(x).

For the function f(x) = 2 \(\ln (x-1)\), you follow these steps but with an extra layer of complexity due to the presence of the natural logarithm. Once you switch x and y, you solve for y using exponential rules to undo the logarithm, arriving at the inverse function f^{-1}(x) = e^{(x/2)}+1. This process is crucial as it forms the foundation for more complex calculus concepts.

Graphing Inverse Functions

Graphing a function alongside its inverse provides visual confirmation of their relationship. To graph these functions, you can use a plotting calculator or software. When graphing f(x) and its inverse f^{-1}(x), you'll find that they're reflections of each other over the line y=x.

This symmetric relationship is best observed by comparing the (x, y) pairs of f(x) with the (y, x) pairs of f^{-1}(x). In our example, the function f(x) = 2 \(\ln (x-1)\) and its inverse f^{-1}(x) = e^{(x/2)}+1 will produce two curves that mirror across the y=x line. This graphical representation is not just visually appealing but also serves as a tool to validate the functions' inverses.

Verification of Inverse Functions

Verification is a critical step to ensure that we have correctly found the inverse function. If f^{-1}(x) is indeed the inverse of f(x), then applying f^{-1} to f(x) should give you back x, and vice versa. Mathematically, this is expressed as f^{-1}(f(x)) = x and f(f^{-1}(x)) = x.

For the function f(x) = 2 \(\ln (x-1)\), verifying its inverse requires substituting the inverse into the original function and confirming that the output matches the input. The algebraic manipulation shuffles the logarithmic and exponential forms until they 'cancel' each other out, leading to the simple x. This verification not only ensures accuracy but also reinforces our understanding of the properties of logarithms and exponentials.

Natural Logarithm ln

The natural logarithm, denoted as ln, is a fundamental concept in calculus and represents the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. It serves as the inverse of the exponential function e^x, meaning ln(e^x) = x and e^(ln(x)) = x.

When working with ln, remember it only accepts positive numbers as input, reflecting that you can’t have a log of a negative number. This property significantly impacts the domain and range of both the logarithmic function and its inverse. Understanding the behavior of ln is crucial when finding and verifying inverse functions involving natural logarithms, as in the example function f(x) = 2 ln(x-1).

Exponential Functions

Exponential functions are powerful tools in mathematics, particularly in modeling growth and decay processes. These functions take the form f(x) = a^x, where a is a positive constant referred to as the base. The number e, as the base of natural logarithms, is often used in exponential functions due to its unique properties in calculus.

Importantly, the inverse of an exponential function is a logarithmic function. In the inverse function f^{-1}(x) = e^{(x/2)} + 1 that we discussed, the base is the constant e and it embodies the exponential quality of rapid growth. Recognizing and working with exponential functions is essential for calculus students, as they're the inverse counterparts of logarithmic functions and appear frequently in real-world applications.