Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-6,\) each set represents a function. a. What is the domain of each function? b. What is the range of each function? c.Is the function one-to-one? $$ \\{(0,8),(2,6),(4,4),(6,2)\\} $$

Step-by-Step Solution

Answer

Domain: \(\{0, 2, 4, 6\}\), Range: \(\{8, 6, 4, 2\}\), Yes, it's one-to-one.

Step 1: Identify the Domain

The domain of a function is the set of all possible input values (x-values). In the given set \( \{(0,8),(2,6),(4,4),(6,2)\} \), the x-values are 0, 2, 4, and 6. Therefore, the domain is \( \{0, 2, 4, 6\} \).

Step 2: Identify the Range

The range of a function is the set of all possible output values (y-values). For the given set \( \{(0,8),(2,6),(4,4),(6,2)\} \), the y-values are 8, 6, 4, and 2. Hence, the range is \( \{8, 6, 4, 2\} \).

Step 3: Determine if the Function is One-to-One

A function is one-to-one if each output value corresponds to exactly one input value, and no two different input values map to the same output value. Here, each y-value in \( \{(0,8),(2,6),(4,4),(6,2)\} \) is unique and corresponds to a unique x-value. Thus, the function is one-to-one.

Key Concepts

One-to-One Function

A one-to-one function is a special type of function that has a unique and distinct property: every output value corresponds to exactly one unique input value. This means there is a one-to-one pairing between inputs (x-values) and outputs (y-values). In simple terms, no two different x-values can produce the same y-value in a one-to-one function.
Why is this important? Because one-to-one functions allow us to determine certain outputs just by knowing the inputs and vice versa, which is a powerful concept in mathematics.
To check if a function is one-to-one, follow these steps:

• Check for unique y-values: Make sure no y-value is repeated for different x-values in the set of ordered pairs.
• Confirm with the function rule: If you have a function rule, you can further validate this property by checking if each y-value results uniquely from the x-value.

In the given set \( \{(0,8),(2,6),(4,4),(6,2)\} \), each y-value is paired with a unique x-value, confirming the function is indeed one-to-one.

Input and Output Values

In any function, particularly one depicted through a set of ordered pairs, identifying the input and output values is crucial to understanding its behavior. These values tell us what is possible within the function's operation.
The input values are also known as the domain. They represent all the possible x-values that you can plug into the function. For every x-value in the domain, there is a corresponding output value.
The output values are known as the range. They consist of all the y-values that you get as a result from plugging x-values into the function.
For example, in the set \( \{(0,8),(2,6),(4,4),(6,2)\} \):

• Domain (inputs): \( \{0, 2, 4, 6\} \)
• Range (outputs): \( \{8, 6, 4, 2\} \)

Understanding these values helps us recognize how the function behaves over its domain.

Function Notation

Function notation offers a standardized way to denote and communicate functions efficiently. This notation helps in identifying the function's rule quickly and understanding how it operates.
A function is often written as \( f(x) \), pronounced "f of x," where:

• \( f \) is the name of the function.
• \( x \) is the input value (from the domain).

Function notation is helpful because it provides a clear formula or rule showing how input values transform into output values. For instance, if you have a function \( f(x) = x^2 + 3 \), it tells you that to get the output, you square the input and then add 3.
In our example set \( \{(0,8),(2,6),(4,4),(6,2)\} \), while we don't have an explicit function rule, writing these pairs as \( f(0)=8 \), \( f(2)=6 \), etc., is useful for understanding the mappings between inputs and outputs. This notation also helps in tracking the transformations within a problem.