Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the functions. $$ f(x)=4-2 x^{3} $$

Step-by-Step Solution

Answer

\( f^{-1}(x) = \sqrt[3]{\frac{4 - x}{2}} \)

Step 1: Write the function in terms of y

Start by writing the function in terms of \( y \); this means replacing \( f(x) \) with \( y \). So we have:\[ y = 4 - 2x^3 \]

Step 2: Swap x and y

To find the inverse of the function, we need to swap the roles of \( x \) and \( y \). This means:\[ x = 4 - 2y^3 \]

Step 3: Solve for y

Now, we need to solve this equation for \( y \) to express the inverse function. First, subtract 4 from both sides:\[ x - 4 = -2y^3 \]Next, multiply both sides by \(-1\):\[ 4 - x = 2y^3 \]Divide both sides by 2:\[ \frac{4 - x}{2} = y^3 \]Finally, take the cube root of both sides:\[ y = \sqrt[3]{\frac{4 - x}{2}} \]

Step 4: Express the inverse function

Now that we have solved for \( y \), we can write the inverse function:\[ f^{-1}(x) = \sqrt[3]{\frac{4 - x}{2}} \]

Key Concepts

Function Notation

Function notation is a way to express the output of a function based on an input, typically using symbols like \( f(x) \). This notation provides a simple yet powerful way to describe mathematical relationships. The expression \( f(x) = 4 - 2x^3 \) tells us that the function named \( f \) takes an input \( x \), processes it using the formula on the right of the equals sign, and gives us a result. Using function notation helps us quickly identify and express the operation applied to the input. It also allows for easy communication of more complex relationships, as each component—like \( x \) and \( f(x) \)—has a clear role:

• \( x \) represents the input value, which can vary.
• \( f(x) \) represents the output or the result of applying the function to \( x \).

This is useful when we need to find specific values or characteristics of a function. In the context of finding inverse functions, function notation provides clarity as we switch roles between the variables, which is crucial for correctly determining the inverse.

Cubic Functions

Cubic functions are polynomial functions where the highest degree of the variable is three. In simple terms, they are characterized by a term in the format of \( x^3 \). The function \( f(x) = 4 - 2x^3 \) is a cubic function because of the \( -2x^3 \) term. Key features of cubic functions include:

• A single real root or three real roots, depending on whether the cubic discriminant is positive, negative, or zero.
• Their distinctive "S" shape curve, which can appear "flatter" or more "stretched" depending on coefficients.
• A change in direction points called inflection points, where the curvature of the graph changes.

Cubic functions are important when finding inverses, as they often involve multi-step processes like solving for \( y \), swapping \( x \) and \( y \), and using operations such as cube roots to isolate variables. Understanding their graph shapes and behavior is crucial, as it provides insight into their increasing or decreasing trends across the domain.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. This task requires a series of logical steps that manipulate the equation to isolate the desired variable on one side of the equation. In the example, starting from \( x = 4 - 2y^3 \), the goal is to solve for \( y \). Here's a concise breakdown of the steps involved:

• Subtract any constants from both sides of the equation to get the variable term alone, such as \( x - 4 = -2y^3 \).
• If necessary, multiply or divide each side by constants to simplify the equation, like dividing by \(-2\) here gives us \( 4 - x = 2y^3 \).
• Apply inverse operations to nonlinear terms—in this case, taking the cube root—resulting in \( y = \sqrt[3]{\frac{4-x}{2}} \).

These techniques not only help in finding inverse functions but are also applicable in broader mathematical contexts. Mastery of solving equations ensures a strong foundation for tackling complex problems, including those involving function inverses such as cubic functions.