Problem 50 - Elementary and Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=(x-9)^{3}\)

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[3]{x} + 9 \).

Step 1: Understand the Function

The function given is a one-to-one function, meaning each output is paired with exactly one input. The function is expressed as \( f(x) = (x - 9)^3 \).

Step 2: Swap Variables

To find the inverse function, we start by swapping \( f(x) \) with \( y \). Thus, our function becomes \( y = (x - 9)^3 \). The goal is to solve for \( x \).

Step 3: Solve for x in terms of y

Start by taking the cube root of both sides to isolate \( x - 9 \): \( \sqrt[3]{y} = x - 9 \).

Step 4: Isolate x

Add 9 to both sides to finally solve for \( x \): \( x = \sqrt[3]{y} + 9 \).

Step 5: Express the Inverse Function

The inverse function \( f^{-1}(x) \) is determined as \( f^{-1}(x) = \sqrt[3]{x} + 9 \).

Key Concepts

One-to-One Functions

A one-to-one function, also known as an injective function, has a unique quality. It pairs every output with a distinct input. In simple terms, no two different inputs will ever produce the same output. This characteristic is critically important when dealing with inverse functions. Since each output corresponds to only one input, it guarantees that inverse functions will exist, and further allows us to reverse the function unambiguously.
Understanding this principle ensures accurate calculation of the inverse.

• Conceptualize the function as a unique pairing between inputs and outputs.
• Verify if it's one-to-one by checking whether different inputs lead to unique outputs.
• If every output maps back to one unique input, you can confidently find the inverse.

Recognizing a function as one-to-one greatly simplifies the effort required to find its inverse.

Solving Equations

To find the inverse of a function, mastering the art of solving equations becomes essential. The given original function was transformed step by step to isolate variables and solve for the desired variable. Here's how that process typically unfolds:
Initially, we rewrite the function to swap dependent and independent variables, using function notation, but that's just the beginning.
First, identify the correct operation needed to isolate the desired variable. For instance, reversing operations like squaring might involve square rooting, while subtraction can be undone with addition. In the example provided, the cube root was used to reverse cubing.

• Determine each step carefully to smoothly transition from one operation to its respective inverse.
• Always perform operations on both sides of the equation to maintain balance.
• Consider each operation and its inverse carefully, one-size-fits-all rules don't apply.

Solving step by step, while being meticulous in each algebraic move, ensures the correct inverse is obtained.

Function Notation

Function notation is compact and tidy, giving clarity and precision to equations and solutions. Having a solid grasp on this notation helps in understanding equations more clearly. In mathematics, function notation typically involves expressions like \( f(x) \), which tells us that \( f \) is a function depending on \( x \).
This becomes even more significant when dealing with inverses. When finding an inverse function, it is represented using notation like \( f^{-1}(x) \). This denotes the inverse of function \( f \). Here are some tips on handling function notation:

• Always denote the original function with \( f(x) \) and the inverse with \( f^{-1}(x) \) after determination.
• Stick to the correct notation to avoid confusion, especially when dealing with inverse functions and interpreting results.
• The notation helps not only in writing less, but also in providing a concise representation of complex mathematical ideas.

Understanding function notation makes mathematical discussions straightforward and is crucial for expressing inverse functions clearly and accurately.