Problem 62 - College Algebra with Modeling and Visualization

Question

Find a symbolic representation for \(f^{-1}(x).\) $$ f(x)=\frac{2}{2-x^{3}} $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \left(\frac{2-2x}{x}\right)^{1/3}.\)

Step 1: Understand the Goal

The goal is to find a symbolic representation for the inverse function, denoted as \(f^{-1}(x)\), of the given function \(f(x) = \frac{2}{2-x^3}\).

Step 2: Swap and Replace

The first step in finding the inverse function is to swap the variables \(x\) and \(y\), where \(y = f(x)\). This gives us \(x = \frac{2}{2 - y^3}\). We'll solve this equation for \(y\) to find \(f^{-1}(x)\).

Step 3: Solve for y

Multiply both sides by \(2 - y^3\) to get rid of the fraction: \[(2-y^3)x = 2.\]Simplify and rearrange to solve for \(y^3\):\[2 - 2x = y^3 x.\]

Step 4: Express y in terms of x

Further isolate \(y^3\):\[y^3 x = 2 - 2x,\] leading to\[y^3 = \frac{2 - 2x}{x}.\]

Step 5: Find the Cube Root

Finally, solve for \(y\) by taking the cube root:\[y = \left(\frac{2 - 2x}{x}\right)^{1/3}.\]This expression is the inverse function \(f^{-1}(x)\).

Step 6: Verify the Inverse (Optional)

To ensure correctness, one could substitute \(f^{-1}(x)\) into the original function \(f(x)\) to check if \(f(f^{-1}(x)) = x\). While optional for this problem, it's a good practice for testing solutions. The composition should yield the identity function.

Key Concepts

Symbolic Representation

When dealing with inverse functions, it's essential to understand how symbolic representation works. In mathematics, a function is often written as a formula or expression, like \( f(x) = \frac{2}{2-x^3} \). This is known as symbolic representation. It describes how to compute function values using algebraic symbols.

For inverse functions, the symbol \( f^{-1}(x) \) is used, indicating the function that "undoes" the effect of \( f(x) \). We are often tasked with finding this inverse formula based on the given expression for \( f(x) \).

To find such a representation:

• Start by setting \( y = f(x) \).
• Swap the roles of \( x \) and \( y \) for the inverse operation, leading to a new expression \( x = g(y) \).
• Solve this transformed equation for \( y \) to find \( f^{-1}(x) \).

Algebraic Manipulation

Algebraic manipulation is a key step in finding inverse functions. It involves rearranging and solving equations to express one variable in terms of another. Going from \( x = \frac{2}{2-y^3} \) to \( y = \left(\frac{2-2x}{x}\right)^{1/3} \) demonstrates several core techniques you need to understand.

First, you need to clear fractions by multiplying both sides of the equation by the denominator. This helps to eliminate complexity and allows more straightforward steps to isolate terms. Next, you should rearrange the equation to obtain terms involving \( y \) on one side, leading you closer to the solution.

Key stages include:

• Multiplying out fractions to simplify expressions.
• Isolating terms by shifting them across the equals sign.
• Further simplifying by collecting like terms or moving them around to isolate \( y^3 \).
• Solving complex expressions by taking cube roots or applying other necessary operations.

By systematically applying these steps, one can solve for \( y \) effectively, thus deriving \( f^{-1}(x) \).

Function Composition

Function composition is a concept where we apply one function to the results of another, written as \( f(g(x)) \) or \( g(f(x)) \). In the context of inverse functions, this becomes particularly useful for verifying the correctness of an inverse.

Once you have \( f(x) \) and its inverse \( f^{-1}(x) \), to verify them, you could check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This process ensures that the inverse function truly "undoes" the operation of the original function.

The steps to verify an inverse function include:

• Composing the original function with the inverse: Substitute \( f^{-1}(x) \) back into \( f(x) \). This should simplify to yield \( x \).
• Composing the inverse function with the original: Substitute \( f(x) \) back into \( f^{-1}(x) \), and this too should simplify to \( x \).

This not only confirms the correctness but also builds confidence in the accuracy of your inverse function manipulation.