Problem 54 - A Graphical Approach to Precalculus with Limits

Question

For each exponential function \(f\), find \(f^{-1}\) analytically and graph \(f\) and \(f^{-1}\) as \(Y_{1}\) and \(Y_{2}\) in the same viewing window. $$f(x)=-e^{x}+6$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \ln(6 - x)\).

Step 1: Understanding the Function

The given exponential function is \( f(x) = -e^{x} + 6 \). The function consists of the exponential function \( e^{x} \) multiplied by \(-1\) and then shifted up by 6 units.

Step 2: Find the Inverse Function

To find the inverse \( f^{-1}(x) \), start by swapping \( x \) and \( y \) in the equation. This gives us \( x = -e^{y} + 6 \). Solve for \( y \) by isolating the exponential term first: \(-e^{y} = x - 6\). Then, solve for \( y \): \( e^{y} = 6 - x \). Finally, take the natural logarithm to isolate \( y \): \( y = \ln(6 - x) \). So, \( f^{-1}(x) = \ln(6 - x) \).

Step 3: Graphing the Functions

The graphs of the functions \( f(x) = -e^{x} + 6 \) and \( f^{-1}(x) = \ln(6 - x) \) should be plotted on the same set of axes. The function \( f(x) \) is an exponential decay (reflected over the \( x \)-axis and shifted 6 units up), and \( f^{-1}(x) \) is a logarithmic function with a horizontal reflection about \( x = 6 \). The line \( y = x \) will act as a line of symmetry for these inverse functions.

Key Concepts

Inverse Functions

To understand inverse functions, let's dive into the basics. An inverse function essentially undoes the action of the original function. If you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), applying \( f \) and then \( f^{-1} \) will get you back to your starting point. Here’s the fundamental relationship:

• If \( y = f(x) \), then \( x = f^{-1}(y) \).
• So \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

For the function \( f(x) = -e^x + 6 \), we find the inverse by swapping \( x \) and \( y \) to get \( x = -e^y + 6 \). Solving for \( y \), you’ll isolate the exponential expression, and finally use the natural logarithm to find \( y = \ln(6 - x) \). Hence, the inverse function is \( f^{-1}(x) = \ln(6 - x) \). Remember, the inverse and the original function will be symmetric along the line \( y = x \). This symmetry is vital when graphing the functions to visually confirm they are true inverses.

Graphing Functions

Graphing functions involves plotting points or curves to visually represent the relationship defined by a function. For exponential functions like \( f(x) = -e^x + 6 \), the curve represents an exponential decay since the exponential part \( e^x \) is negative. The graph is flipped over the x-axis due to the negative sign, creating a curve that decreases and levels out as \( x \) increases, shifted upward by 6 units.
When graphing the inverse, in this case \( f^{-1}(x) = \ln(6 - x) \), you'll observe a logarithmic curve. Unlike exponential decay, this curve increases as \( x \) approaches 6 from the left, pivoting vertically down as \( x \) decreases past zero.
While plotting both graphs, it's useful to incorporate the line \( y = x \) to explicitly show the symmetry between a function and its inverse. This line is where their intersections would confirm their inverse relationship. Utilizing graphing software or a coordinate plane is recommended for precise representation and analysis of these varying curves.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a crucial mathematical function, especially when dealing with exponential functions. It is based on the constant \( e \) (approximately 2.718), an important base for natural logarithms. The natural log function \( \ln(x) \) essentially finds the exponent required to obtain \( x \) from \( e \). In simpler terms, if \( e^y = x \), then \( y = \ln(x) \).

• Natural logarithms are particularly essential for working with inverse functions of exponential forms.
• They help in "undoing" the effect of the exponential function, thus making them key in solving equations involving \( e^x \).

In the given problem, applying the natural logarithm allows us to isolate \( y \) when finding the inverse, leading from \( e^y = 6 - x \) to \( y = \ln(6 - x) \). Understanding this concept helps in graphing and comprehending how exponential decay and growth translate visually, thus reinforcing the grasp of inverse relationships.