Problem 16 - Elementary and Intermediate Algebra

Question

Complete each solution. Find the inverse of \(f(x)=\sqrt[3]{x}+2\) (SOLUTION CAN'T COPY)

Step-by-Step Solution

Answer

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

Step 1: Express the function in terms of y

Start by replacing the function notation \( f(x) \) with \( y \). This gives us the equation:\[ y = \sqrt[3]{x} + 2 \]

Step 2: Isolate the root term

To find the inverse, solve for \( x \). First, isolate the cube root term:Subtract 2 from both sides:\[ y - 2 = \sqrt[3]{x} \]

Step 3: Eliminate the cube root

To eliminate the cube root, cube both sides of the equation:\[ (y - 2)^3 = x \]

Step 4: Express x in terms of y

Now that you have \( x \) isolated, express \( x \) in terms of \( y \):\[ x = (y - 2)^3 \]

Step 5: Write the inverse function

The inverse of the function is found by interchanging \( x \) and \( y \), and expressing \( y \) as a function of \( x \):\[ f^{-1}(x) = (x - 2)^3 \]

Key Concepts

Understanding Function Notation

Function notation is a concise way to express the relationship between a set of inputs and outputs in mathematics. Instead of writing out equations repeatedly, function notation allows us to define a function, say \(f\), with \(x\) as the input variable, and easily reference it across different contexts.

• In the equation \( f(x) = \sqrt[3]{x} + 2 \), \( f(x) \) is the function notation, signaling that \( f \) is a function of \( x \).
• This indicates that when you plug a number into \( x \), the right side of the equation will yield the corresponding output for the function \( f \).
• Replacing \( f(x) \) with \( y \) helps simplify processing the equation for further manipulations, as seen in the initial step of finding the inverse: \( y = \sqrt[3]{x} + 2 \).

Function notation is essential when working with functions, especially for transformations and inverse calculations, because it provides a clear and flexible framework for manipulating expressions and understanding relationships between variables.

The Cube Root and Its Role in Functions

The cube root, represented as \( \sqrt[3]{x} \), is the value that, when raised to the power of three, equals \( x \). This operation is pivotal in simplifying and understanding equations, particularly when examining functions.

• For the function \( f(x) = \sqrt[3]{x} + 2 \), the cube root \( \sqrt[3]{x} \) transforms any input \( x \) by finding a number that, when cubed, gives \( x \).
• To "undo" the cube root during the inverse function process, we cube both sides of the manipulated equation, effectively reversing the cube root operation.
• Cubing both sides of the equation \( y - 2 = \sqrt[3]{x} \) gives \( (y - 2)^3 = x \), isolating \( x \).

Understanding cube roots is crucial for handling nonlinear transformations and finding inverse functions, where applying the opposite operation resolves the equation for the input or domain.

Solving Equations to Determine Inverses

To find the inverse of a function, solving equations is a fundamental process. This task involves expressing the original function in terms of a new variable and then reversing the operations.

• Beginning with \( y = \sqrt[3]{x} + 2 \), we perform operations sequentially to isolate \( x \).
• Subtracting 2 from both sides, \( y - 2 = \sqrt[3]{x} \), starts the process of isolating the variable of interest.
• Next, cubing both sides eliminates the cube root, resulting in \( (y - 2)^3 = x \), making \( x \) the subject of the formula.
• Finally, swapping \( x \) and \( y \), as done in find an inverse, yields the inverse function: \( f^{-1}(x) = (x - 2)^3 \).

Solving equations is integral to determining inverses as it systematically removes constraints, allowing you to express one variable concerning another. This method is central to understanding how one navigates between inputs and outputs in function transformations.