Problem 38 - 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. $$ f(x)=\frac{x}{3}-\frac{1}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

We are given the function \( f(x) = \frac{x}{3} - \frac{1}{3} \). Our task is to find its inverse.

Step 2: Replace f(x) with y

To find the inverse, start by replacing \( f(x) \) with \( y \). Thus, the equation becomes \( y = \frac{x}{3} - \frac{1}{3} \).

Step 3: Solve for x in terms of y

Rearrange the equation \( y = \frac{x}{3} - \frac{1}{3} \) to solve for \( x \):1. Add \( \frac{1}{3} \) to both sides: \( y + \frac{1}{3} = \frac{x}{3} \).2. Multiply both sides by 3 to isolate \( x \): \( 3(y + \frac{1}{3}) = x \).

Step 4: Simplify the expression

Simplify the expression for \( x \):1. Distribute the 3: \( 3y + 1 = x \). So, \( x = 3y + 1 \).

Step 5: Replace x with f^-1(x)

Since we have expressed \( x \) in terms of \( y \), replace \( x \) with \( f^{-1}(x) \), and \( y \) with \( x \) in the equation \( x = 3y + 1 \). Thus, \( f^{-1}(x) = 3x + 1 \).

Key Concepts

One-to-One Functions

In mathematics, understanding one-to-one functions is essential for grasping the concept of inverse functions. A one-to-one function, also known as an injective function, is a type of function where each element of the range is mapped by a unique element of the domain. In simpler terms, different inputs lead to different outputs. This property ensures that the function can be reversed, leading to the concept of an inverse function.

**Why is One-to-One Important?**

• This unique mapping is crucial because only one-to-one functions have inverses that are also functions.
• If a function is not one-to-one, some outputs might map back to more than one input when reversed, which wouldn't work as a proper function.

Recognizing if a function is one-to-one can often be done through the Horizontal Line Test, where a horizontal line should not intersect the graph of the function at more than one point. This assures us that the function can safely have an inverse that is also a function.

Solving Equations

Solving equations is a fundamental skill in finding inverse functions. It's the process of finding the unknown variable that makes the equation true. When solving equations to find an inverse, you typically express one variable in terms of another.

In the exercise, we want to go from the function form to a rearranged form where we express the original input variable in terms of the output variable. Let's break it down:

• Start with an equation, typically in the form of \( y = f(x) \).
• Manipulate the equation to solve for \( x \) in terms of \( y \).
• This often involves simple algebraic operations like addition, subtraction, and multiplication.

For example, with the function \( y = \frac{x}{3} - \frac{1}{3} \), we add \( \frac{1}{3} \) to both sides and then multiply by 3 to get \( x \) by itself. This simplifies to \( x = 3y + 1 \), ready to transition into its inverse.

Function Notation

Function notation is a compact way to represent functions and their inverses. It allows us to easily express and work with functions mathematically. When discussing functions and their inverses, we'll see notations like \( f(x) \) and \( f^{-1}(x) \).

**What Does Function Notation Tell Us?**

• \( f(x) \) represents the function output for a given input \( x \).
• \( f^{-1}(x) \) is used to denote the inverse function, which does the opposite: it takes the output and returns the original input.

This notation is especially useful because it communicates which function we are talking about easily and effectively, without writing out longer descriptions. For instance, using \( f^{-1}(x) = 3x + 1 \) shows how we can calculate the original input from any output \( x \) of \( f(x) \), making it clear and organized.