Problem 53 - 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. \(f(x)=\frac{x^{7}}{2}\)

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = (2x)^{1/7} \).

Step 1: Understand the One-to-One Function

The given function is \( f(x) = \frac{x^7}{2} \). A one-to-one function maps each input to a unique output, meaning every \( y \)-value has exactly one corresponding \( x \) value. To find the inverse, we will swap the roles of \( x \) and \( y \).

Step 2: Set Up the Equation for the Inverse

Start by letting \( y = f(x) \), so \( y = \frac{x^7}{2} \). To find the inverse, we want to express \( x \) in terms of \( y \). So, replace \( f(x) \) with \( y \): \( y = \frac{x^7}{2} \).

Step 3: Swap and Solve for the Original Variable

Swap \( x \) and \( y \) to find the inverse: \( x = \frac{y^7}{2} \). Now solve for \( y \) by multiplying both sides by 2: \[ 2x = y^7 \]. Then, take the seventh root of both sides to solve for \( y \): \[ y = (2x)^{1/7} \].

Step 4: Write the Inverse Function in Notation

Now express the inverse function using \( f^{-1}(x) \): \( f^{-1}(x) = (2x)^{1/7} \).

Key Concepts

One-to-One Functions

In mathematics, a one-to-one function, or injective function, is one where each input maps to a distinct output. Simply put, no two different inputs will produce the same output in a one-to-one function. This is an essential property when discussing inverse functions because only one-to-one functions have inverses that are also functions.

To understand this better:

• Consider the function as a machine: different inputs always yield different outputs.
• If you draw a horizontal line across the graph of a one-to-one function, it will intersect the graph at most once.

When finding an inverse, recognizing that the function is one-to-one assures us that we can reverse the process. In our original exercise, knowing that the function \(f(x) = \frac{x^7}{2}\) is one-to-one gives us the green light to find an actual inverse that will behave like a function itself.

Function Notation

Function notation is a fancy name for a simple idea - it is a way to indicate the function and its inputs and outputs clearly. When you see \(f(x)\), it's like saying "the function of \(x\)." Functions can take in numbers, transform them according to their rules, and then output a result.

Breaking it down into smaller parts:

• \(f\) is the name of the function.
• \(x\) is the input variable.
• \(f(x)\) represents the output after applying the function rule to \(x\).

Understanding this notation is key to working with functions and their inverses. The inverse notation \(f^{-1}(x)\) tells us that this is not a typical power of \(f\) but rather a new function that undoes the original \(f\). This helps track which process you are applying at any given time, especially when switching between a function and its inverse.

Solving Equations

Solving equations is at the heart of finding inverse functions. When finding an inverse, we essentially reverse the operations of the original function step by step. For instance, in the exercise, we started with \(y = \frac{x^7}{2}\) and needed to express \(x\) in terms of \(y\).

This process generally involves:

• Swapping the roles of \(x\) and \(y\), which flips the input-output relationship.
• Using algebraic techniques, such as multiplying or taking roots, to isolate the variable \(y\).

For example, after swapping in our exercise, we multiplied both sides by 2 to get \(2x = y^7\). Taking the seventh root gave us \(y = (2x)^{1/7}\). Finally, we express this result using inverse function notation: \(f^{-1}(x) = (2x)^{1/7}\). Solving equations allows us to backtrack through a function's operations, revealing its inverse.