Problem 20 - Intermediate Algebra

Question

Determine whether each function is one-to-one. $$ f(x)=|x| $$

Step-by-Step Solution

Answer

The function \( f(x) = |x| \) is not one-to-one.

Step 1: Understanding One-to-One Functions

A function is one-to-one if each element of the range is paired with exactly one element of the domain. In simpler terms, if different inputs (x-values) always produce different outputs (f(x)-values), then the function is one-to-one.

Step 2: Analyze the Given Function

The function given is \( f(x) = |x| \), which denotes the absolute value function. This function converts all negative inputs to positive, while positive inputs (including zero) remain unchanged.

Step 3: Test for Distinct Inputs Producing Distinct Outputs

Check if there are distinct inputs that could produce the same output. For example, for \( x = 3 \) and \( x = -3 \), both inputs provide the output \( f(x) = 3 \).

Step 4: Conclusion from Input-Output Analysis

Since two different inputs, \( x = 3 \) and \( x = -3 \), lead to the same value in the range (\( f(x) = 3 \)), \( f(x) = |x| \) does not meet the one-to-one condition.

Key Concepts

Absolute Value Function

The absolute value function, represented as \( f(x) = |x| \), is a mathematical function that transforms any input into its non-negative form. Whether the input is positive or negative, the function outputs the magnitude of the number, effectively disregarding the sign. For instance, \(|3| = 3\) and \(|-3| = 3\).

This property of the absolute value function means that it maps both negative and positive inputs of the same magnitude to the same output. This characteristic plays a crucial role when analyzing whether \(f(x) = |x| \) is a one-to-one function. Here, it's essential to remember that for a function to be completely characterized as one-to-one, there shouldn’t be different inputs that result in the exact same output.

Function Analysis

When performing function analysis on \(f(x) = |x| \), we pay close attention to how the function behaves with different inputs. Our goal is to understand the relationships formed between inputs and outputs.

In function analysis, we assess whether a function like \(f(x) = |x| \) satisfies particular properties necessary for classification, such as being one-to-one. This includes determining if every output is uniquely connected to an input. For the absolute value function, when you input 3, the output is 3. When you input -3, the output is also 3. This repetition of output values from distinct inputs hints at the non-uniqueness required for a function to be labeled as one-to-one.

Input-Output Analysis

Input-output analysis is a method used to scrutinize the connection between the values we put into a function and the values we obtain as a result. In the context of the absolute value function \(f(x) = |x| \), this kind of analysis helps us determine whether respective varying inputs can result in identical outputs.

Consider inputs of -3 and 3. Both yield the same output of 3, which indicates these outputs are not unique for each input, disqualifying \(f(x) = |x| \) from being one-to-one. What we are essentially doing is mapping the inputs (domain) to the outputs (range) and seeing if there is a case where different inputs map to the same output. If such cases exist, as they do here, the function cannot be regarded as one-to-one. Thus, through input-output analysis, we confirm the conclusion that \(f(x) = |x| \) is not one-to-one.