Problem 71 - Precalculus Mathematics for Calculus

Question

Find the inverse function of \(f\). $$f(x)=2^{2 x}$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{1}{2} \log_2(x)\).

Step 1: Understand the function

The given function is \(f(x) = 2^{2x}\). This is an exponential function where \(2x\) is the exponent of the base \(2\). Our goal is to find the inverse of this function.

Step 2: Replace f(x) with y

To find the inverse of the function, start by substituting the function \(f(x)\) with \(y\). This gives us the equation: \(y = 2^{2x}\).

Step 3: Swap x and y

To find the inverse, swap \(x\) and \(y\) in the equation. The equation \(y = 2^{2x}\) becomes \(x = 2^{2y}\).

Step 4: Solve for y in terms of x

We need to solve \(x = 2^{2y}\) for \(y\). Start by taking the logarithm of both sides, specifically base 2. This gives: \(\log_2(x) = 2y\).

Step 5: Isolate y

To isolate \(y\), divide both sides of the equation by 2, resulting in \(y = \frac{1}{2} \log_2(x)\).

Step 6: Express the inverse function

Since \(y = \frac{1}{2} \log_2(x)\), the inverse function \(f^{-1}(x)\) is given by: \(f^{-1}(x) = \frac{1}{2} \log_2(x)\). This is the final solution.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In our exercise, the function \( f(x) = 2^{2x} \) demonstrates this, with base 2 and the exponent being \( 2x \). The primary characteristic of exponential functions is their rapid growth or decay, depending on whether the base is greater than or less than 1. In our case, since the base is 2, \( f(x) \) exhibits growth.

Understanding this expansion is crucial, as it allows us to appreciate the mechanism of the inverse. In real-world scenarios, exponential functions describe phenomena like population growth, radioactive decay, and interest calculations. Their steep curves make them unique yet solveable, as we find ways to lower the complexity like taking logarithms to reverse their effects.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It helps us 'undo' an exponentiation. In the problem, to find the inverse of \( f(x) = 2^{2x} \), we used logarithms. The equation \( x = 2^{2y} \) was transformed by applying logarithms, specifically \( \log_2 \), to isolate \( y \). This method is beneficial often because logarithms turn the multiplication or power of variables into addition or division.

In practical terms, logarithmic scales are used where we need to model a wide range of values, like in measuring acidity (pH), sound intensity, or stellar brightness. The property of logarithmic functions that makes them incredibly useful is how they naturally reverse exponential processes, making them indispensable in both theoretical and practical mathematics.

Function Operations

Function operations involve manipulating one or more functions to produce a desired function or solve for certain values. In this exercise, we conducted function operations by swapping the variables \( x \) and \( y \) and then solving for the new arrangement. This swap is essential to derive an inverse function, indicating the process of finding which input corresponds to an output of a function.

This swap and solve technique highlights the intricacy of inverse function operations. By understanding these, you gain insights into backtracking through processes, a crucial skill across disciplines, from solving simple equations to decrypting messages. Function operations help clarify how changes in input values reflect in outputs, and inversely, how outputs reveal their originating inputs.