Problem 83 - Precalculus : Building Concepts and Connections

Question

Give an example of an odd function that is not one-to-one.

Step-by-Step Solution

Answer

A function that is both odd and not one-to-one is the sinusoidal function \(f(x) = \sin(x)\).

Step 1: Understand Properties

First, understand the properties of an odd function and a not one-to-one function. An odd function f(x) must satisfy f(-x) = -f(x) for all x in the domain of the function. A not one-to-one function means, there are at least two distinct values in the domain that map to the same value in the range.

Step 2: Select Function

One common function that satisfies both these characteristics is a sine function. We can use \( f(x) = \sin(x) \) as our example as it holds both the required properties.

Step 3: Confirm Odd Function Properties

For a function to be odd, it needs to satisfy the condition f(-x) = -f(x). For our chosen function f(x) = \(\sin(x)\), we can confirm this as \(\sin(-x) = -\sin(x)\). Thus, the property of an odd function is satisfied.

Step 4: Confirm not One-to-One Function Properties

Now, check the requirement for a not one-to-one function, which is that at least two distinct elements from the domain must map to similar elements in the range. For our chosen function, sin(x), this is satisfied as sin(x) = sin(x + 2π,) for all x in the domain of the function. Thus, there are different values in the domain which point to the same value in range(e.g., sin(0) = sin(2π)). Therefore, our chosen function sin(x) is indeed not one-to-one.

Key Concepts

Understanding One-to-One Functions

A **one-to-one function** is a concept where each input value maps uniquely to a distinct output value. This means if you have two different input values, they should not map to the same output value. Mathematically, a function \( f(x) \) is one-to-one if:

• For any \( a \) and \( b \) in the domain, if \( f(a) = f(b) \), then \( a = b \).

In simpler terms, if two inputs give you the same output, they must be the same input.
Visualizing this is simple with a horizontal line test: If a horizontal line intersects the graph of the function more than once, the function is not one-to-one.
This distinct characteristic ensures no two different inputs will result in the same output, making it an important property in many mathematical applications.

Exploring the Sine Function

The **sine function**, represented as \( f(x) = \sin(x) \), is a familiar wave-like function within trigonometry. It is periodic, meaning it repeats its values in a regular cycle, with a period of \( 2\pi \).
Key elements of the sine function include:

• It oscillates between -1 and 1.
• It has a period of \( 2\pi \), meaning \( \sin(x) = \sin(x + 2\pi k) \) for any integer \( k \).

Because of this periodic nature, the sine function inherently cannot be one-to-one over its entire domain.
It repeats its values at regular intervals, ensuring that different inputs, such as 0 and \( 2\pi \), lead to the same outcome (e.g., \( \sin(0) = \sin(2\pi) = 0 \)).
Thus, while it's a classic and widely-used function, its wave-like repetition means it isn't one-to-one.

Understanding Function Properties

When exploring **function properties**, it’s valuable to recognize characteristics like oddness and how they interact with one-to-one behavior.
An **odd function** satisfies: \( f(-x) = -f(x) \). This means its graph is symmetric about the origin.
The sine function, \( f(x) = \sin(x) \), is an excellent example of an odd function:

• It fulfills \( \sin(-x) = -\sin(x) \), showing its symmetry around the origin.

Additionally, understanding the property of being not one-to-one requires identifying if distinct domain values can yield the same output.
For \( \sin(x) \), this holds as shown by \( \sin(x) = \sin(x + 2\pi) \).
Thus, grasping these properties not only distinguishes the nature of functions but also highlights how specific characteristics like being odd can support their classification.