Problem 25 - Elementary and Intermediate Algebra

Question

Determine whether each function is one-to-one. \(\\{(1,1),(2,1),(3,1),(4,1)\\}\)

Step-by-Step Solution

Answer

The function is not one-to-one.

Step 1: Understanding a One-to-One Function

A function is considered one-to-one if each output (or y-value) is paired with exactly one unique input (or x-value). In other words, no two different inputs should be mapped to the same output.

Step 2: Analyzing the Given Function

Examine the given set of ordered pairs: \((1,1), (2,1), (3,1), (4,1)\). Here, the output or y-value is 1 for each input or x-value.

Step 3: Identifying Repeated Output Values

Notice that the output y-value of 1 is repeated for different inputs: 1, 2, 3, and 4. This means for the same output of 1, there are multiple x-values.

Step 4: Conclusion on One-to-One Property

Since multiple x-values (1, 2, 3, 4) correspond to the same y-value (1), the function is not one-to-one. A one-to-one function cannot have repeating y-values for different x-values.

Key Concepts

Understanding a Set of Ordered Pairs

In mathematics, a set of ordered pairs is a collection of elements grouped together with a specific first and second position. It is usually represented as

• \((x, y)\), where \(x\) is the first element known as the x-value, and \(y\) is the second element known as the y-value.

Ordered pairs are crucial in determining relationships between two different quantities, such as inputs and outputs in a function. To understand a function as a set of ordered pairs, imagine it as a collection of pairs like

• \((1, 1), (2, 1), (3, 1), (4, 1)\).

Each pair illustrates how one input (x-value) is related to an output (y-value). This understanding is vital for analyzing properties like one-to-one functions.

Exploring X-Value

The x-value, also known as the input, is the first element in an ordered pair. When examining if a function is one-to-one, it's essential to observe the x-value associated with each output or y-value. For example, in the set of ordered pairs

• \((1, 1), (2, 1), (3, 1), (4, 1)\),the x-values are 1, 2, 3, and 4.

This signifies that these different inputs are given to the function to produce outputs. If each y-value corresponds to a unique x-value, the function is one-to-one. Paying attention to how these x-values are paired with y-values helps determine the characteristics of functions.

The Role of Y-Value

The y-value is the output in the set of ordered pairs. When assessing whether a function is one-to-one, the y-value plays a crucial role in the analysis. Consider the set

• \((1, 1), (2, 1), (3, 1), (4, 1)\).Here, the y-value of 1 is repeated for different x-values: 1, 2, 3, and 4.

For a function to be one-to-one, each y-value must be paired uniquely with one x-value. In this example, multiple x-values map to the same y-value. This repetition indicates the function is not one-to-one. Observing the y-value and its relationships helps in drawing conclusions about the function's properties.

Function Analysis for One-to-One Property

Function analysis involves evaluating how inputs are related to outputs to determine various characteristics of the function, such as whether it is one-to-one. A function is classified as one-to-one if it uniquely pairs each x-value with a single y-value, meaning no two x-values share the same y-value.
In the given set of ordered pairs

• \((1, 1), (2, 1), (3, 1), (4, 1)\),multiple x-values (1, 2, 3, and 4) yield the same y-value (1).

This mapping indicates the absence of a one-to-one relationship due to repeated y-values for different x-values. Designating functions as one-to-one allows for precise calculations and solutions in various mathematical contexts, reinforcing the significance of function analysis.