Problem 538 - Precalculus

Question

Find the inverse function \(f^{-1}\) for the logarithmic function \(f(x)=0.25 \cdot \log _{2}\left(x^{3}+1\right)\).

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(y) = \sqrt[3]{2^{4y} - 1} \).

Step 1: Understand the Problem

We want to find the inverse function of the given logarithmic function. The function is \( f(x) = 0.25 \cdot \log_2(x^3 + 1) \). The goal is to solve for \( x \) in terms of \( y \) where \( y = f(x) \).

Step 2: Write the Function in Terms of y

First, express the function with \( y \) in place of \( f(x) \):\[ y = 0.25 \cdot \log_2(x^3 + 1) \]

Step 3: Isolate the Logarithmic Term

To isolate the logarithmic part, divide both sides of the equation by 0.25:\[ \log_2(x^3 + 1) = 4y \]

Step 4: Remove the Logarithm

Use the property of logarithms to rewrite the equation in exponential form:\[ x^3 + 1 = 2^{4y} \]

Step 5: Solve for x in terms of y

Subtract 1 from both sides to isolate the \( x^3 \) term:\[ x^3 = 2^{4y} - 1 \]Then solve for \( x \) by taking the cube root:\[ x = \sqrt[3]{2^{4y} - 1} \]

Step 6: Write the Inverse Function

The inverse function is found as \( f^{-1}(y) = \sqrt[3]{2^{4y} - 1} \). This represents the function that reverses the effect of \( f(x) \).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They help us find the exponent that a base must be raised to in order to get a certain number. In simpler terms, if you know the result of an exponential equation and want to find the exponent that was used, you use a logarithm.
For example:

• If we have a number like 8 and a base of 2, we can express this as:
  \( \log_2(8) = x \)
• Here, you ask: 2 to what power gives me 8? The answer is 3, so \( \log_2(8) = 3 \).

Logarithms can be written in different bases, but two of the most common are base 10, denoted as \( \log(x) \) or \( \log_{10}(x) \), and base \( e \), the natural logarithm denoted as \( \ln(x) \). In our exercise, the function used base 2 as the logarithmic base. The logarithmic function \( f(x) = 0.25 \cdot \log_2(x^3 + 1) \) evaluates the logarithm of \( (x^3 + 1) \) and then scales it by a factor of 0.25.

Exponential Equations

An exponential equation is any equation where a variable appears in the exponent. These are the opposite of logarithmic functions because they involve finding a number by raising a base to a certain power. For example, if you start with \( 2^3 \), this equals 8 because 2 multiplied by itself three times equals 8. This is a basic exponential equation.
In our specific exercise, we transformed the equation from a logarithmic form to an exponential form to find the inverse. Here's a key step:

• After isolating the logarithmic part, \( \log_2(x^3 + 1) = 4y \), you transitioned it to the exponential form:
  \( x^3 + 1 = 2^{4y} \).

This means you are converting the equation using the property that \( b^{\log_b(a)} = a \). Such transformations allow us to solve for the variable in terms of the other, which is crucial for finding inverse functions.

Function Notation

Function notation is a way to express a function's rule that relates inputs to outputs. Functions are usually written as \( f(x) \), where \( f \) is the name of the function and \( x \) represents an input to the function. This notation helps us understand what operation is being performed.
When you find the inverse of a function, as in our problem, you want to reverse what the original function does. For instance, if \( f(x) = 0.25 \cdot \log_2(x^3 + 1) \), the inverse function \( f^{-1}(y) \) reverses this process. If \( y = f(x) \), then \( x = f^{-1}(y) \). In this case, the inverse function was found to be \( f^{-1}(y) = \sqrt[3]{2^{4y} - 1} \). This represents how the output \( y \) from the original function maps back to the input \( x \).
Understanding function notation is key to grasping these relationships and effectively managing transformations between original functions and their inverses.