Problem 14 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. $$ g(x)=|x| $$

Step-by-Step Solution

Answer

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

Step 1: Understand One-to-One Functions

A function is one-to-one (injective) if each element of the range is paired with exactly one element of the domain. In other words, no two different inputs have the same output.

Step 2: Analyze the Function Expression

The given function is \( g(x) = |x| \). This function takes any real number \( x \) and gives its absolute value. Absolute value, \( |x| \), is always non-negative and depends only on the magnitude of \( x \), not its sign.

Step 3: Check Function Behavior on Positive and Negative Inputs

Examine values such as \( g(2) \) and \( g(-2) \). Both are equal to \( 2 \). This duplicate output from different inputs (\( 2 \) and \( -2 \)) suggests that \( g(x) \) is not one-to-one, since different inputs can give the same output.

Step 4: Conclude the Analysis

Since there exist different inputs that have the same output for \( g(x) = |x| \), the function is not one-to-one. For a function to be one-to-one, each output must correspond to one unique input.

Key Concepts

Function Analysis

When analyzing functions, it's essential to explore how inputs relate to outputs. A function is a rule that assigns each input exactly one output. In function analysis, we study these relationships to understand properties like domain, range, and behavior. There are different types of functions depending on their characteristics:

• One-to-One (Injective): Each output is related to a unique input.
• Onto (Surjective): Every possible output is an actual output of the function.
• Bijective: Both one-to-one and onto.

An essential part of function analysis is identifying whether a function is one-to-one. This involves checking that no two distinct inputs map to the same output. If any output has more than one input, the function is not one-to-one. In the given example with \( g(x) = |x| \), we specifically need to consider the absolute value, which influences this determination.

Absolute Value Function

The absolute value function is a special type of mathematical function that returns the non-negative value of a given number. Represented as \( |x| \), the absolute value simply measures the distance of a number from zero on the number line. A few key characteristics of absolute value functions include:

• The output is always non-negative.
• \( |x| = x \) if \( x \geq 0 \), and \( |x| = -x \) if \( x < 0 \).
• The graph is a "V" shape centered at the origin (0,0).

Absolute value functions often have different outputs for positive and negative inputs, which can lead to the same absolute result. For instance, the function \( g(x) = |x| \) will output 2 for both \( x = 2 \) and \( x = -2 \). This symmetry is why it fails the one-to-one test, as multiple inputs produce the same output. The understanding of the absolute value is crucial here as it dictates why \( g(x) = |x| \) results in overlapping outputs for different inputs.

Injective Function

An injective function, also known as a one-to-one function, is characterized by its unique mapping of inputs to outputs. Specifically, if a function is injective, no two different inputs will map to the same output.To check if a function is injective:

• Pick two different values \( x_1 \) and \( x_2 \) in the domain such that \( x_1 eq x_2 \).
• Substitute both values into the function and ensure \( f(x_1) eq f(x_2) \).

For example, in determining whether \( g(x) = |x| \) is injective, note how \( |x| \) maps both positive \( x \) and its negative counterpart to the same non-negative output. For instance, both \( g(1) \) and \( g(-1) \) are 1. Therefore, because \( |x| \) assigns the same output to these different inputs, it is not injective. Understanding injectivity is key, as it allows us to discern whether the function has distinct or overlapping outputs for distinct inputs.