Question

Give an example of a function whose domain is the set of real numbers and that is neither increasing nor decreasing on its domain, but is one-to-one. [Hint: Use a piecewise-defined function.]

Short answer

A piecewise function such as \( f(x) = \begin{cases} x, & x \text{ is even} \ -x, & x \text{ is odd} \end{cases} \) meets all the requirements.

Step-by-step solution

Understand the Requirements

Identify that the function must satisfy three conditions: it must be defined for all real numbers, it must be neither increasing nor decreasing throughout its domain, and it must be one-to-one (injective).

Construct a Piecewise-Defined Function

Design a function that behaves differently on different intervals of its domain. Ensure it satisfies all the given conditions.

Define the Function

Here is an example of such a function: \[ f(x) = \begin{cases} x, & x \text{ is even} \ -x, & x \text{ is odd} \end{cases} \]For this function, observe that the function is defined everywhere on the set of real numbers, not increasing or decreasing consistently, but remains one-to-one.

Verify the Function Meets All Conditions

Check three conditions:1. **Domain:** The function is defined for all real numbers, as specified in the piecewise definition.2. **Not Increasing/Decreasing:** The function does not consistently increase or decrease since it alternates based on the parity of the input.3. **One-to-One:** Each input value results in a unique output value, preventing any repetition in output for different inputs.

Key concepts

Domain of a Function

The domain of a function is the set of all possible input values (typically represented by 'x') for which the function is defined. For many functions, this includes all real numbers, but some functions have restricted domains due to factors like division by zero or square roots of negative numbers.

• For the function in our exercise, the domain is the set of all real numbers.
• This means that no matter what real number you plug into the function, you will get a result.

In the piecewise-defined function provided as the solution, the domain includes all real numbers because it specifies behavior for every real number input. This thorough definition is pivotal for satisfying the domain condition.

One-to-One Function

A one-to-one function (also known as injective) is a function where each input (x) has a unique output (f(x)). This means no two different inputs can map to the same output.

• In the given function, regardless of whether x is even or odd, each x maps to a unique (f(x)).
• For example, if x=2 (even), f(x)=2, and if x=3 (odd), f(x)=-3.

Ensuring that each input results in a distinct output is key to proving that a function is one-to-one. You can test this by assuming f(a) = f(b) and showing that a must equal b for the given function.

Increasing and Decreasing Functions

A function is termed increasing if, for any two numbers in its domain, a larger input gives a larger output. Conversely, a function is decreasing if a larger input results in a smaller output.
However, some functions neither wholly increase nor decrease. They may go up and down depending on the segment of the domain examined.

• In our exercise, the function fluctuates between positive and negative values depending on whether x is even or odd.
• This means the function does not consistently increase or decrease across its domain, fulfilling the required condition.

Identifying sections of a piecewise function is crucial to understand its overall behavior and proving it is neither increasing nor decreasing.