Problem 1 - Algebra 2 and Trigonometry

Question

If the domain of the function \(f(x)=|3-x|\) is the set of real numbers less than \(3,\) is the function one-to-one? Explain why or why not.

Step-by-Step Solution

Answer

Yes, the function is one-to-one for the domain \(x < 3\), as it behaves linearly and each input maps to a unique output.

Step 1: Understanding the Function

The given function is \(f(x) = |3 - x|\). The absolute value can change the behavior of the function around \(x = 3\).

Step 2: Define the Domain

The domain specified is the set of all real numbers less than 3, or \(x < 3\). This means our function operates only on these values.

Step 3: Examine Function Behavior in Domain

For \(x < 3\), the function simplifies to \(f(x) = 3 - x\). Therefore, within the domain \(x < 3\), the function behaves as \(f(x) = 3 - x\).

Step 4: Check for One-to-One Property

A function is one-to-one if different inputs produce different outputs. Let's assume \(f(a) = f(b)\) for some \(a, b < 3\), then \(3 - a = 3 - b\). Solving, we get \(a = b\), which confirms that different inputs lead to different outputs.

Key Concepts

Domain and Range

Understanding the domain and range of a function is essential for analyzing its behavior. The **domain** consists of all possible input values. For the function \(f(x) = |3-x|\), the domain is explicitly given as all real numbers less than 3, meaning \(x < 3\). This restriction focuses the function on a specific part of the real number line, leading to more precise behavior analysis.

The **range** of a function consists of all possible output values. Since the function \(f(x) = |3-x|\) simplifies to \(f(x) = 3-x\) within its restricted domain, the outputs will vary linearly from values just below 3 to positive, yet progressively smaller numbers as \(x\) approaches 3 from the negative side. Thus, the range of the function is \(f(x) > 0\) when \(x < 3\). Understanding these concepts clarifies what the function can output given its inputs.

Absolute Value Functions

Absolute value functions can drastically change how a function behaves, especially near points of critical change like \(x = 3\) in our example. The absolute value \(|3-x|\) ensures that output values never become negative. Its role is to reflect any negative input result back into its positive equivalent.

For \(x < 3\), the function simplifies significantly since the values inside the absolute value bracket don't need to be flipped. Therefore, \(f(x) = 3-x\) across the entire considered domain. It becomes straightforward to analyze how changes in \(x\) affect \(f(x)\) without the absolute value interfering. Such clarifications help prevent misunderstandings about how absolute value might alter function behavior across different domains.

Function Behavior Analysis

To understand the behavior of \(f(x) = |3-x|\) in its given domain (\(x < 3\)), it is essential to determine whether it is one-to-one. A function is said to be **one-to-one** if every distinct input produces a distinct output. This property can be validated by assuming two different inputs yield the same output, thereby leading to a contradiction unless both inputs are the same.

For \(f(x) = 3 - x\), if \(f(a) = f(b)\) for some \(a, b < 3\), then \(3-a = 3-b\). Simplifying this, we find \(a = b\), which confirms that different inputs indeed lead to different outputs within the specified domain. As a linear function without any resurgence of previous output values, \(f(x)\) is thus one-to-one over its defined domain. Understanding this behavior allows students to see why functions hold or break specific linearity or bijective properties.