Problem 10 - 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 the Definition of a One-to-One Function

A function is one-to-one (injective) if and only if every element of the function's range corresponds with exactly one element of the domain. This means that different elements in the domain map to different elements in the range.

Step 2: Analyze the Given Function

The given function is \( g(x) = |x| \). This means the function takes any real number \( x \) and outputs its absolute value, which is always non-negative.

Step 3: Test Key Criteria for One-to-Oneness

To determine if the function is one-to-one, check if two different inputs can lead to the same output. Consider \( x = -1 \) and \( x = 1 \): both give \( g(-1) = |-1| = 1 \) and \( g(1) = |1| = 1 \).

Step 4: Conclude Based on Observations

Since both \( x = -1 \) and \( x = 1 \) produce the same value (1) as the output, it shows that \( g(x) \) does not give unique outputs for every unique input. Therefore, \( g(x) = |x| \) is not one-to-one.

Key Concepts

Understanding Injective Functions

To grasp the concept of an injective function, or a one-to-one function, it's important to realize that it maps distinct inputs to distinct outputs. This means for every point in the domain, there is a unique partner in the range. If any two different values in the domain map to the same value in the range, the function is not injective. Visualize it as a strict, one-on-one relationship where no two different members of the domain share the same member of the range. This unique mapping can ensure that the function passes the "horizontal line test," where no horizontal line intersects the graph in more than one point. Remember, injection is all about contrasting each input's uniqueness and ensuring distinct results for distinct inputs.

Exploring the Absolute Value Function

The absolute value function, represented as \( g(x) = |x| \), is a well-known mathematical function that outputs the distance of a number from zero on the number line, making it always non-negative. Simply put, it gives the magnitude of a number without considering its sign.

• If \( x \) is positive or zero, \( |x| = x \).
• If \( x \) is negative, \( |x| = -x \).

As a result, the graph of the absolute value function forms a "V" shape, mapping both positive and negative inputs to positive outputs. Such properties make understanding whether \( g(x) = |x| \) is injective straightforward, as it reflects that multiple inputs could return the same output, thus challenging the one-to-one nature.

Delving into Function Analysis

Function analysis is a critical tool in mathematics that helps examine the behavior and properties of functions systematically. To analyze a function thoroughly, consider aspects like domain, range, and specific points that illustrate key behaviors.

• Domain: For the absolute value function, the entire set of real numbers can be considered a domain since it can take any real number.
• Range: It provides only non-negative results, so the range is non-negative real numbers.
• Critical Test: When analyzing for injectiveness, as shown in the exercise, you need to identify whether any two different inputs yield the same output, as with \( x = -1 \) and \( x = 1 \).

In essence, function analysis helps decipher intricate details and uncover vital characteristics, aiding in forming solid conclusions about the nature of the function in question, just as determining the non-injectiveness of \( g(x) = |x| \) using critical tests.