Problem 59 - 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. $$f(x)=-10^{x}+4$$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \log_{10}(-x + 4) \), defined for \( x < 4 \).

Step 1: Understand the Function

Identify the type of function given. The function is \( f(x) = -10^x + 4 \). This is an exponential function with a vertical shift of 4 units upwards and a reflection across the x-axis.

Step 2: Find the Inverse

To find the inverse of the function \( f(x) = -10^x + 4 \), start by setting \( y = f(x) \). This gives us \( y = -10^x + 4 \). Solve for \( x \) in terms of \( y \): 1. Subtract 4 from both sides: \( y - 4 = -10^x \).2. Multiply both sides by \( -1 \) to get \( -y + 4 = 10^x \).3. Take the logarithm base 10 of both sides: \( \log_{10}(-y + 4) = x \).Thus, the inverse function is \( f^{-1}(x) = \log_{10}(-x + 4) \).

Step 3: Check the Domain

Check the domain of the inverse function, \( f^{-1}(x) = \log_{10}(-x + 4) \). The argument of the logarithm, \(-x + 4\), must be positive: \(-x + 4 > 0\) which simplifies to \(x < 4\). Hence, the domain of \( f^{-1} \) is \( x < 4 \).

Step 4: Graph the Functions

Graph both \( f(x) = -10^x + 4 \) and \( f^{-1}(x) = \log_{10}(-x + 4) \) in the same viewing window. 1. The function \( f(x) \) decreases from 4 to negative infinity. 2. The inverse function \( f^{-1}(x) \) is defined for \( x < 4 \) and will show a logarithmic decrease. Ensure the graph reflects the correct domain restrictions.

Key Concepts

Inverse Functions

Finding the inverse function of an exponential function is an interesting task. For the function \( f(x) = -10^x + 4 \), the inverse \( f^{-1}(x) \) essentially tells us how to reverse the effect of the original function and find the initial input from a given output.

• First, denote the function as \( y = -10^x + 4 \).
• To switch back variables, solve for \( x \) in terms of \( y \). This involves rearranging the equation: subtract \( 4 \) from both sides to get \( y - 4 = -10^x \).
• Multiply by \( -1 \) to simplify: \( -y + 4 = 10^x \).
• Take the logarithm base 10: \( \log_{10}(-y + 4) = x \), resulting in the inverse function \( f^{-1}(x) = \log_{10}(-x + 4) \).

This process illustrates the principle of inverse functions—a function that "undoes" another function. The key here is manipulation of the equation to express \( x \) in terms of \( y \), effectively "reversing" the input-output relationship.

Domains of Functions

Domains are crucial since they define all possible input values for a function. For functions involving logarithms, such as \( f^{-1}(x) = \log_{10}(-x + 4) \), only positive arguments are permissible under the logarithmic function.When identifying the domain of \( f^{-1}(x) \):

• The expression inside \( \log_{10}(-x + 4) \) must be positive, so we set \( -x + 4 > 0 \).
• This simplifies to \( x < 4 \), establishing the domain as all real numbers smaller than 4.

For the original function \( f(x) = -10^x + 4 \):

• The domain is all real numbers because exponential functions are defined everywhere.

Thus, carefully checking domains ensures that our calculations remain valid and our graphs accurately represent the functions involved.

Graphing Functions

Graphing both the original function and its inverse can provide a visual understanding of their relationship. When plotting \( f(x) = -10^x + 4 \) and \( f^{-1}(x) = \log_{10}(-x + 4) \), we can observe how each curve behaves.A few key graphing tips:

• The graph of \( f(x) \) will show an exponential decrease with a horizontal reflection and a vertical shift, starting from \( y = 4 \) and decreasing as \( x \) increases.
• For \( f^{-1}(x) \), only consider \( x < 4 \), where the graph depicts a decrease with a distinctly logarithmic shape. It approaches negative infinity as \( x \) approaches 4 from the left.
• Both graphs should show symmetry over the line \( y = x \) if plotted together on the same axis.

By understanding their graphs together, one can see more clearly how inverse operations truly "flip" the input and output of functions and comprehend their domains and ranges more intuitively.