Problem 20 - Elementary and 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 because different inputs can produce the same output.

Step 1: Understand the Definition of One-to-One Functions

A function is considered one-to-one if each output value is associated with a unique input value. In other words, no two different input values should produce the same output.

Step 2: Analyze the Function

Consider the function \( f(x) = |x| \). The absolute value function takes any real number \( x \) and returns its non-negative value, which means it outputs the same value for both positive and negative inputs. For example, \( f(1) = |1| = 1 \) and \( f(-1) = |-1| = 1 \). The function gives the same output for different inputs, \( 1 \) and \( -1 \).

Step 3: Conclusion from Analysis

Since \( f(x) = |x| \) maps different inputs to the same output, it does not satisfy the condition of a one-to-one function. Therefore, \( f(x) = |x| \) is not a one-to-one function.

Key Concepts

Understanding the Absolute Value Function

The absolute value function, denoted as \( f(x) = |x| \), is a mathematical operation that takes any real number \( x \) and returns its non-negative value. In simpler terms, this function removes any negative sign, effectively measuring the 'distance' of a number from zero on the number line. For instance:

• If \( x = 3 \), then \( f(x) = |3| = 3 \).
• If \( x = -4 \), then \( f(x) = |-4| = 4 \).

The absolute value function is symmetric around the y-axis, meaning that both positive and negative inputs result in the same output value. This inherent property makes it a bit challenging when considering whether it is a one-to-one function. The symmetry indicates that multiple inputs can produce the same result, which is a key aspect to explore when determining characteristics like being one-to-one.

Exploring Function Analysis

Function analysis involves understanding the behavior and properties of functions, which can include determining its domain, range, and whether it's one-to-one. A function is one-to-one if for every unique input there is a unique output. This means no two different input values can map to the same output value.To analyze the function \( f(x) = |x| \), consider its outputs for different inputs:

• Both \( f(1) \) and \( f(-1) \) result in 1, demonstrating that different inputs map to the same output.
• This symmetry shows that \( f(x) \) doesn't satisfy the condition for being one-to-one.

This property can help in determining whether further transformations or adaptations of such functions can be one-to-one, or if additional constraints can make them injective, which is another term used for this property.

Applying Mathematical Reasoning

Mathematical reasoning allows us to logically deduce the properties of functions based on structured analysis and definitions. By systematically applying such reasoning to the absolute value function, we can conclude non-intuitive properties with confidence.When determining if \( f(x) = |x| \) is one-to-one, mathematical reasoning requires us to verify if our function satisfies the one-to-one criterion across its entire domain. Because \( f(x) \) produces the same result for distinct inputs, both positive and negative versions of a number, we use logical deductions:

• If \( x_1 = -x_2 \) and \( x_1 e x_2 \), yet \( |x_1| = |x_2| \), then \( f(x) \) fails to be one-to-one.
• Reasoning through these examples helps strengthen our understanding that different approaches or strategies might be needed to achieve the one-to-one property in similar functions, such as restricting the domain to non-negative inputs only.

This reasoning framework provides a basis for more advanced analysis and exploration in mathematics, enhancing the robust understanding of complex functions and their behavior.