Problem 58 - 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. $$f(x)=\left(\frac{1}{2}\right)^{x}-5$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \log_{\frac{1}{2}}(x + 5) \).

Step 1: Understand the function

We are given the exponential function \( f(x) = \left( \frac{1}{2} \right)^x - 5 \). This function represents a transformation of the basic exponential function by scaling it with a base of \( \frac{1}{2} \) and shifting it downward by 5 units.

Step 2: Set y = f(x)

To find the inverse, set \( y = f(x) \). Thus, we have \( y = \left( \frac{1}{2} \right)^x - 5 \).

Step 3: Solve for x in terms of y

Add 5 to both sides of the equation: \( y + 5 = \left( \frac{1}{2} \right)^x \).

Step 4: Remove the exponent

To solve the equation for \( x \), we use logarithms. Take the logarithm base \( \frac{1}{2} \) of both sides: \( x = \log_{\frac{1}{2}}(y + 5) \).

Step 5: Interpret the inverse

The function \( f^{-1}(x) \) is defined as \( f^{-1}(x) = \log_{\frac{1}{2}}(x + 5) \). This function represents the inverse to \( f(x) \).

Step 6: Graph f(x) and f^{-1}(x)

Plot \( f(x) = \left( \frac{1}{2} \right)^x - 5 \) and its inverse \( f^{-1}(x) = \log_{\frac{1}{2}}(x + 5) \) on the same axes. The graph of \( f(x) \) is exponentially decreasing due to the base \( \frac{1}{2} \), and the graph of \( f^{-1}(x) \) is the reflection of \( f(x) \) across the line \( y = x \).

Key Concepts

Inverse Functions

Inverse functions are a fascinating topic in mathematics. Essentially, an inverse function reverses the operations of the original function. If you input a value into the function and then feed the result into its inverse, you’ll get back to your original input. This is useful in many real-world scenarios where we need to reverse a process.
To find the inverse of a function analytically means to determine a formula that describes this reverse relationship. For instance, in an exponential function like \( f(x) = \left( \frac{1}{2} \right)^x - 5 \), finding the inverse involves switching the variables so \( y = f(x) \) becomes \( x = f^{-1}(y) \). The key step involves isolating \( x \) in terms of \( y \). This often requires algebraic manipulation, just like we’ve done by adding 5 to both sides and then using logarithms to solve for \( x \).
Thus, after cleansing the function of its transformations—like the shift in our example—the inverse function \( f^{-1}(x) = \log_{\frac{1}{2}}(x + 5) \) reflects this reverse operation.

Graph Interpretation

Graph interpretation of functions and their inverses provide visual insights into mathematical equations. When graphing both a function and its inverse in the same coordinate plane, a key feature to observe is the property of reflection over the line \( y = x \).
In the given example, plotting \( f(x) = \left( \frac{1}{2} \right)^x - 5 \) will show an exponential decay because of its base, \( \frac{1}{2} \). As we move along the x-axis, the function values decrease. With the inverse \( f^{-1}(x) = \log_{\frac{1}{2}}(x + 5) \), graphing it shows how it traces the mirrored path along the line \( y = x \).
Graphing helps in understanding how the values relate between \( f \) and \( f^{-1} \). For example:

• The y-intercept of \( f(x) \) becomes the x-intercept for \( f^{-1}(x) \), and vice versa.
• Features such as asymptotes in \( f(x) \), like the horizontal asymptote at \( y = -5 \), become vertical asymptotes in \( f^{-1}(x) \).

Overall, graphing both functions enriches comprehension by offering visual relation and comparison.

Logarithms

When dealing with exponential functions, logarithms become an essential tool. Logarithms are the inverse operations of exponentiation. They help us solve equations where the unknown variable is in the exponent, such as finding inverses of exponential functions.
For the function \( f(x) = \left( \frac{1}{2} \right)^x - 5 \), finding its inverse required taking the logarithm. Specifically, we used the same base as the exponential, resulting in the expression \( x = \log_{\frac{1}{2}}(y + 5) \). This allowed for isolating \( x \) resulting in the inverse function, \( f^{-1}(x) \).
The logarithmic function itself has a few crucial properties:

• It is the inverse of an exponential function. Thus, \( \log_b(b^x) = x \).
• Logarithms transform multiplicative relationships into additive ones, simplifying complex exponential equations.

Logarithms are not just tools for solving mathematical problems; they appear in real-world applications, including sound intensity (decibels), earthquake intensity (Richter scale), and even financial models like compound interest.