Problem 60 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is \( f(x) = -e^{x} + 6 \). This is an exponential function where \( e^{x} \) is the exponential term. It has been reflected and shifted upwards by 6.

Step 2: Set y in Terms of x

To find the inverse function \( f^{-1}(x) \), first replace \( f(x) \) with \( y \): \( y = -e^{x} + 6 \). We want to solve for \( x \) in terms of \( y \).

Step 3: Isolate the Exponential Term

Rearrange the equation to isolate the exponential term: \( -e^{x} = y - 6 \). Multiply both sides by \(-1\) to get \( e^{x} = 6 - y \).

Step 4: Solve for x Using Natural Logarithm

Take the natural logarithm of both sides to solve for \( x \): \( x = \ln(6 - y) \). This gives \( x \) in terms of \( y \).

Step 5: Express x in Terms of y for Inverse

Switch \( x \) and \( y \) to express the inverse function: \( y = \ln(6 - x) \). Thus, \( f^{-1}(x) = \ln(6 - x) \).

Step 6: Graph Both Functions

Graph \( f(x) = -e^{x} + 6 \) and \( f^{-1}(x) = \ln(6 - x) \) in the same window. The graph of \( f(x) \) is a vertical reflection and shift of the exponential curve, while \( f^{-1}(x) \) is a logarithmic function mirrored over the line \( y = x \).

Key Concepts

Exponential Functions

Exponential functions have the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. In the exercise, the function \( f(x) = -e^{x} + 6 \) is slightly altered from the basic form. Here, the function is multiplied by -1, which reflects it over the x-axis. This creates a downward opening curve. Additionally, it is shifted upward by 6 units, moving the entire graph higher on the y-axis.

Some characteristics of exponential functions include:

• Rapid growth or decay, depending on whether the base is greater or less than 1.
• A horizontal asymptote, which in this case is the line \( y = 6 \) due to the shift.

Understanding the transformation can help in predicting the behavior of exponential functions under various manipulations.

Graphing

Graphing exponential and inverse functions can provide a clear visual understanding of their relationship. In this exercise, graphing \( f(x) = -e^{x} + 6 \) and its inverse \( f^{-1}(x) = \ln(6 - x) \) helps to show the symmetry about the line \( y = x \).

Here are some tips when plotting these functions:

• For \( f(x) = -e^{x} + 6 \), start by graphing the basic \( e^{x} \) curve, then reflect it below the x-axis and shift it up 6 units.
• For \( f^{-1}(x) = \ln(6 - x) \), note that it will only be defined for \( x < 6 \), showing an undefined point at 6.
• Plot important points such as intercepts and asymptotes to guide the curve.

Graphing helps reaffirm the theoretical understanding, enhancing learning through visualization.

Natural Logarithms

Natural logarithms, denoted as \( \ln(x) \), are the inverse of exponential functions with base \( e \). In the problem, using natural logarithms helps solve the inverse for the given exponential function.

When isolating \( x \) in terms of \( y \), we arrive at the equation \( e^{x} = 6 - y \). To solve for \( x \), applying the natural log on both sides gives us \( x = \ln(6 - y) \).

Key points about natural logarithms include:

• They reverse the exponential operation. If \( e^x = a \), then \( \ln(a) = x \).
• Logarithmic functions grow very slowly, notably much slower than exponential functions.
• The domain of \( \ln(x) \) is for \( x > 0 \), imposing restrictions like \( x < 6 \) in this context.

Understanding natural logarithms leads to better intuition about exponential relationships and their inverses.