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 \(f\left(f^{-1}(x)\right)=x\). $$ f(x)=e^{4 x-1} $$

Short answer

The inverse of the function \(f(x) = e^{4x-1}\) is \(f^{-1}(x) = \frac{1}{4}ln(x) + \frac{1}{4}\). Graphs of the functions should show the inverse function as a reflection of the original over the line y=x. The functions \(f(x)\) and \(f^{-1}(x)\) satisfy the properties \(f\left(f^{-1}(x)\right) = x\) and \(f^{-1}(f(x)) = x\), confirming that they are indeed inverses.

Step-by-step solution

Finding the Inverse

The inverse of a function interchanges the roles of y and x. Hence, start by replacing \(f(x)\) with y, \(y = e^{4x-1}\). Given the nature of exponential functions, apply the natural logarithm (ln) to both sides to 'undo' the exponent, obtaining \(ln(y) = 4x-1\). Now, isolate x to express the function in inverse notation \(f^{-1}(x)\). Therefore, \(f^{-1}(x) = \frac{1}{4}ln(x) + \frac{1}{4}\).

Graphing the Function and its Inverse

Use a graphing utility to graph the original function \(f(x) = e^{4x-1}\) and its inverse \(f^{-1}(x) = \frac{1}{4}ln(x) + \frac{1}{4}\). The graph of \(f^{-1}(x)\) should appear as a reflection of the graph of \(f(x)\) over the line y=x since they are inverse functions.

Verifying the Function Properties

Verify whether \(f\left(f^{-1}(x)\right) = x\) and \(f^{-1}(f(x)) = x\). Substituting \(f^{-1}(x)\) into the function \(f(x)\) or vice versa should result in x. For \(f\left(f^{-1}(x)\right)=x, replace \(f^{-1}(x)\) in \(f(x)\) and simplify to get x. Similarly, for \(f^{-1}(f(x))=x, replace \(f(x)\) in \(f^{-1}(x)\) and simplify to get x. If both result in x, then the functions are indeed inverses.

Key concepts

Exponential Functions

Exponential functions, typically represented as f(x) = a^x, where a is a positive constant, are fundamental in calculus for modeling growth and decay scenarios. The base of the natural exponential function, e, which is approximately 2.71828, has unique properties that make it particularly valuable in mathematical and real-world applications.

For instance, the function f(x) = e^{4x-1} from our exercise represents exponential growth. The constant multiplier in the exponent, in this case, 4, affects the rate of change of the function, causing it to grow quickly as x increases. Conversely, as x approaches negative infinity, the function approaches zero but never reaches it.

Natural Logarithm

The natural logarithm is the inverse operation of the natural exponential function. For any positive number x, the natural logarithm of x, denoted as ln(x), is the power to which e must be raised to obtain x. So, if y = e^x, then ln(y) = x.

This relationship is crucial when finding the inverse of exponential functions, as shown in our step-by-step solution, where applying ln to both sides of the equation y = e^{4x-1} helps to 'isolate' x and find the inverse function. Understanding the natural logarithm function is key to working with exponential relationships in calculus.

Graphing Utility

Graphing utilities are invaluable tools in calculus for visualizing the behavior and relationships of functions. They enable us to plot complex equations with ease and can provide a visual confirmation of algebraic work, such as checking the accuracy of an inverse function.

To utilize a graphing utility for functions such as f(x) = e^{4x-1} and its inverse, one would typically enter these functions into the tool and adjust the viewing window to capture the key features of the graphs. The graphical representation should show that the inverse is indeed a mirror image of the original function across the line y=x. This reflection is a characteristic property of inverse functions.

Function Verification

Verifying whether two functions are inverses of each other is a critical aspect of calculus. As the step-by-step solution illustrates, we confirm that f^{-1}(f(x)) = x and f(f^{-1}(x)) = x for the functions in question. This verification process is done by substituting one function into the other and simplifying.

If the end result is x in both cases, the functions are proven to be true inverses. Such verification ensures that the functions' compositions undo each other, which is the defining property of inverse functions. This concept is not only essential in mathematics but is also applied in real-world situations where reversing a process is needed.