Problem 41 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=5-4 x^{3} $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{-\frac{x-5}{4}}\).

Step 1: Replace Function Notation with Variables

Rewrite the function \(f(x) = 5 - 4x^3\) using \(y\) instead of \(f(x)\). This gives the equation: \(y = 5 - 4x^3\).

Step 2: Switch x and y

Swap \(x\) and \(y\) in the equation to prepare for finding the inverse. This gives: \(x = 5 - 4y^3\).

Step 3: Solve for y

Rearrange the equation \(x = 5 - 4y^3\) to solve for \(y\):1. Subtract 5 from both sides: \(x - 5 = -4y^3\).2. Divide both sides by -4: \(-\frac{x-5}{4} = y^3\).3. Take the cube root of both sides: \(y = \sqrt[3]{-\frac{x-5}{4}}\).

Step 4: Write the Inverse Function

The inverse function is obtained from the solved \(y\). Thus, the inverse function is: \(f^{-1}(x) = \sqrt[3]{-\frac{x-5}{4}}\).

Key Concepts

Function Notation

Understanding function notation is the first step toward grasping how functions work and how to find their inverses. In mathematics, a function is often represented using a specific notation. For example, the function \( f(x) = 5 - 4x^3 \) tells you how the output is calculated using the input \( x \).
This notation \( f(x) \) is simply a way of saying "a function of \( x \)." It's a placeholder that makes it easier to work with functions in equations.

• Rewriting the function with \( y \) instead of \( f(x) \) can make the math more straightforward. This means transforming \( f(x) = 5 - 4x^3 \) to \( y = 5 - 4x^3 \).
• This substitution allows you to focus on the mathematical relationship without worrying about the specific function label.

Next, we proceed to swap variables to prepare for getting the inverse function.

Swap Variables

After converting the function notation into an equation with \( y \), the next step in finding an inverse function is to swap the roles of the variables. In the context of inverse functions, this means switching \( x \) and \( y \).

This swapping is crucial because the inverse function is essentially switching the input and output. By swapping, we reframe the equation from \( y = 5 - 4x^3 \) to \( x = 5 - 4y^3 \).

• Think of it as exchanging the dependent and independent variables to indicate that you're reversing the process of the function.
• Swapping variables helps reveal how the original inputs become outputs in the inverse, and vice versa.

Once variables are swapped, the goal shifts to solving for \( y \).

Solve for y

With the variables swapped and the equation \( x = 5 - 4y^3 \) established, it's time to solve for \( y \). Solving for \( y \) involves isolating \( y \) on one side of the equation, which requires careful algebraic manipulation.
Here are the steps involved:

• First, subtract 5 from both sides, leading to \( x - 5 = -4y^3 \).
• Next, divide both sides by \(-4\) to simplify, resulting in \(-\frac{x-5}{4} = y^3 \).
• Finally, apply the cube root to both sides to isolate \( y \), giving \( y = \sqrt[3]{-\frac{x-5}{4}} \).

Solving for \( y \) is essentially unraveling how the original function suggests \( x \) and translating it back into terms of \( y \).
This process lets us express \( y \) explicitly as the output for any given \( x \).
After this, we'll write the inverse function using the solved \( y \).

Cube Root

Taking the cube root is an important algebraic procedure used to solve for \( y \) in equations involving cubes. The cube root "undoes" the cubing process, allowing you to find the original value before it was raised to the third power.

In the equation \(-\frac{x-5}{4} = y^3\), the cube root is essential to isolate \( y \).

• Applying the cube root function to both sides of the equation reverses the power of three, simplifying \( y^3 \) back to \( y \).
• This step allows the equation to become \( y = \sqrt[3]{-\frac{x-5}{4}} \), thus clearly depicting the transformed relationship in terms of the input \( x \).

Understanding cube roots and recognizing when to apply them is crucial when working with cubic relationships and thus finding inverse functions effectively.