Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

We need to find the inverse of the function \( f(x) = 4 - x^3 \). The inverse function, \( f^{-1}(x) \), will reverse the roles of \( x \) and \( y \). Generally, to find the inverse, we swap \( x \) and \( y \) in the equation.

Step 2: Swap Variables

Replace \( f(x) \) with \( y \), so we have \( y = 4 - x^3 \). Swap the variables \( x \) and \( y \), leading to the equation \( x = 4 - y^3 \).

Step 3: Solve for y

Rearrange the equation \( x = 4 - y^3 \) to express \( y \) in terms of \( x \). Start by isolating \( y^3 \) on one side: \( y^3 = 4 - x \).

Step 4: Cube Root Both Sides

Take the cube root of both sides to solve for \( y \), resulting in \( y = \sqrt[3]{4-x} \). This expression represents the inverse function.

Step 5: Write the Inverse Function

Now that we have determined \( y \) in terms of \( x \), the inverse function is \( f^{-1}(x) = \sqrt[3]{4-x} \).

Key Concepts

Function Operations

Functions are equations that describe a relationship between two variables, typically referred to as x (input) and y (output). Function operations can involve performing various algebraic manipulations to transform or combine functions. Understanding function operations is essential, as they allow us to modify functions, find inverses, and solve equations.

To find the inverse of a function, we perform a series of operations that essentially reverse the original function's process. This involves meticulously switching the roles of the output and input variables and undertaking algebraic manipulations to isolate the desired variables. For the function given, we need to swap and solve to express x in terms of y, and vice versa, which leads us to the inverse function.

Solving Equations

Solving equations is a fundamental aspect of mathematics and is crucial when working with functions or finding their inverses. It involves manipulating an equation to find the value of a variable.

In our exercise, after swapping variables, we ended up with the equation \( x = 4 - y^3 \). To solve for y, which represents our inverse function's output, we performed operations to isolate y. This is achieved by moving terms across the equation and manipulating it to solve for the variable explicitly.

• Start by isolating the term with \( y \): \( y^3 = 4 - x \).
• Finally, y can be expressed as \( y = \sqrt[3]{4-x} \).

Solving such equations involves understanding how to move and manipulate parts of the equation correctly to solve for a particular variable.

Cube Root

The cube root operation is essential in finding the inverse of a cubic equation. The cube root of a number, \( n \), is a value that, when multiplied by itself twice, gives \( n \). For example, the cube root of 8 is 2 since \( 2 \times 2 \times 2 = 8 \).

When solving for the inverse function, the term \( y^3 = 4 - x \) arises, and to isolate \( y \), we need to apply the cube root to both sides. This means we are looking for a number \( y \) such that \( y^3 \) equals \( 4 - x \). Performing the cube root operation allows us to express \( y \) as \( y = \sqrt[3]{4-x} \).

Cube roots are important in various fields of science and engineering, helping to simplify and solve equations involving cubic terms.

Swap Variables

Swapping variables is a critical step in finding inverse functions. The act of exchanging variables helps reverse the function's effects, allowing us to express the original dependent variable as an independent variable and vice versa.

In the given exercise, we start with \( y = 4 - x^3 \), representing our function \( f(x) \). To find the inverse, we need to swap x and y, resulting in \( x = 4 - y^3 \). This step is crucial as it begins the inversion process by switching the roles of input and output.

• Original function: \( y = 4 - x^3 \)
• Swapped function: \( x = 4 - y^3 \)

After swapping, further algebraic manipulations are needed to solve for y in terms of x, finalizing the inverse function. Swapping variables unravels the original function's design, leading us to its inverse.