Question

In Exercises \(25-32,\) use parametric graphing to graph \(f, f^{-1},\) and \(y=x\) $$f(x)=e^{x}$$

Short answer

The inverse function of \(f(x) = e^x\) is \(f^{-1}(x) = \ln x\). The graphs of these functions are reflections of each other over the line \(y = x\), confirming they are indeed inverse of each other.

Step-by-step solution

Find the Inverse Function

The function given in the exercise is \(f(x)=e^{x}\). The inverse of an exponential function is a logarithm. Therefore, the inverse function of \(f(x)\) would be \(f^{-1}(x)=\ln x\).

Plot the Graph of the Given Function, Its Inverse and Y = X

Use parametric graphing to sketch the graphs of the function \(f(x) = e^x\), its inverse function \(f^{-1}(x) = \ln x\), and the line \(y = x\). The graph of \(f(x) = e^x\) starts from (0,1) and grows exponentially on the right. The graph of \(f^{-1}(x) = \ln x\) starts from (1,0) and grows logarithmically on the top. The line \(y = x\) is a straight line passing through the origin with slope 1.

Analyze the Graphs

Analyze the obtained graphs. The graph of \(f(x) = e^x\) and its inverse function \(f^{-1}(x) = \ln x\) are reflections of each other over the line y = x. This is indicative of the fact that the functions are indeed inverses of each other.

Key concepts

Exponential Functions

Exponential functions, such as the one presented in the exercise, are of great importance in both mathematics and real-world applications. An exponential function is generally expressed as \( f(x) = a^x \), where '\( a \)' is a constant base and '\( x \)' is the exponent. In our specific example, the base is Euler's number '\( e \)', which is approximately 2.71828.

The function \( f(x) = e^x \) shows how values grow at a rate proportional to their current value, which makes exponential functions model scenarios like population growth, radioactive decay, and compound interest. When graphed, the exponential curve starts from a point (0,1), because any number to the power of zero is one, and then it increases rapidly without ever touching the x-axis, asymptotically approaching zero as \( x \) approaches negative infinity.

Plotting Exponential Functions

Visualization is key for understanding the behavior of exponential functions. You usually see a steeply increasing curve for positive values of \( x \), while for negative \( x \), the curve approaches the x-axis gradually but never reaches zero. It's this unique 'J-shaped' curve that distinguishes exponential growth from linear or polynomial growth.

Inverse Functions

Inverse functions are a fundamental concept in mathematics that reverse the effect of the original function. Given a function \( f(x) \), the inverse function, denoted as \( f^{-1}(x) \), is the function that, when composed with \( f \), yields the identity function, meaning \( f(f^{-1}(x)) = x \).

For the exercise at hand, \( f(x) = e^x \) is an exponential function, and its inverse is the logarithmic function \( f^{-1}(x) = \text{ln}(x) \). Finding an inverse requires that the original function be one-to-one, where each element of the range corresponds to exactly one element of the domain. Graphically, the inverse is the reflection of the original function over the line \( y=x \).

Finding Inverses

To find the inverse function, one typically swaps the roles of \( x \) and \( y \) and then solves for \( y \). For exponential functions, this process naturally leads to logarithmic functions as their inverses, often requiring a good understanding of the properties of logarithms to manipulate and solve the equations correctly.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They are denoted as \( f(x) = \text{ln}(x) \) when the base is '\( e \)', Euler's number. Logarithms answer the question: 'To what exponent must we raise the base to obtain a given number?'. So, \( \text{ln}(x) \) asks for the power that '\( e \)' must be raised to to get \( x \).

These functions are crucial in diverse fields, such as mathematics, science, and engineering, as they are used to measure the intensity of earthquakes (Richter scale), sound (decibels), and even the acidity of solutions (pH scale).

Characteristics of Logarithmic Functions

A common characteristic of logarithmic functions is their logarithmic curve, starting sharply from negative infinity at \( x = 0 \) and gradually increasing to infinity as \( x \) increases. This is the mirror effect of exponential growth when considering logarithms as inverse functions. Plotting logarithmic functions requires an understanding that they are not defined for negative values or zero, as these do not correspond to outputs of exponential functions with a positive base.