Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the relation represents a function. \(\\{(3,4),(4,5),(5,6)\\}\)

Step-by-Step Solution

Answer

Yes, it represents a function because each input has a unique output.

Step 1: Understand the Definition of a Function

A relation is a function if every input (first element of an ordered pair) is associated with exactly one output (second element of an ordered pair). In other words, no input value should map to more than one output value.

Step 2: Identify the Inputs and Outputs

List the pairs in the given relation: \((3,4), (4,5), (5,6)\). Identify the inputs, which are the first elements: 3, 4, and 5. Determine the outputs, which are the second elements: 4, 5, and 6.

Step 3: Check for Unique Outputs

For each input, check if there is exactly one output. Here, input 3 maps to 4, input 4 maps to 5, and input 5 maps to 6. Each input has only one output.

Step 4: Conclusion Based on Definition

Since each input from the relation \((3,4), (4,5), (5,6)\) maps to a unique output, the given relation satisfies the definition of a function.

Key Concepts

Relations and Functions

In algebra, a **relation** is a set of ordered pairs. Each pair consists of two elements: an input and an output. Relations can be represented in many ways: through sets, tables, mappings, or graphs. However, not every relation is called a function.

A **function** is a specific type of relation. For a relation to be a function, each input must correspond to exactly one output. It means that no input value is associated with more than one output value. Think of it like a vending machine: for each button (input) you press, you expect to get exactly one type of snack (output). If pushing a button would give you different snacks each time, that vending machine would not "function" properly!

Functions are essential in algebra because they provide consistent outcomes. When dealing with functions, you can be sure that the relationship between inputs and outputs will be reliable every time you encounter the same input.

Input-Output Pairs

Understanding **input-output pairs** is crucial when working with functions in algebra. Each ordered pair in a relation represents an input followed by its corresponding output. In the example relation \(\{(3,4),(4,5),(5,6)\}\):

• The input of the first pair is 3, resulting in an output of 4.
• The input of the second pair is 4, resulting in an output of 5.
• The input of the third pair is 5, resulting in an output of 6.

For a relation to be a function, each input should produce one, consistent output. Consider a scenario where you enter the number 3 into a machine and it returns the number 4. If you input 3 tomorrow, and instead get the number 9, then this machine would not reflect a function since the output isn't consistent for the same input.

Think of input-output pairs like a map. The input is your starting point, and the output is your destination. A good map (like a function) will always guide you to your intended destination without deviation.

Function Definition

The **definition of a function** is a fundamental concept in mathematics. It establishes the rule that distinguishes functions from general relations. A function is defined as a set of ordered pairs where no two distinct pairs have the same first element (input) but different second elements (outputs).

In simpler terms, a function ensures that each input is linked to one and only one output. This unique pairing is what makes functions predictable and reliable. Using the vending machine analogy again: each selection button should dispense a specific snack every time, without any surprises.

When examining a set of ordered pairs to determine if it qualifies as a function, check if each input in the set has only one corresponding output. For example, in the relation \(\{(3,4),(4,5),(5,6)\}\), every input (3, 4, and 5) is paired with one and only one output, thus fulfilling the criteria of a function. This consistency is the hallmark of a true function in algebra.