Problem 80 - College Algebra with Modeling and Visualization

Question

Determine if \(S\) is a function. $$ S=\\{(a, 2),(a, 3),(b, 5),(-b, 7)\\} $$

Step-by-Step Solution

Answer

S is not a function because input 'a' maps to two different outputs.

Step 1: Understand the Definition of a Function

A set is a function if each input (first component of an ordered pair) is related to exactly one output (second component of an ordered pair). This means no input value can map to more than one output value.

Step 2: Identify the Input and Output Pairs in Set S

Consider the given set of ordered pairs \( S=\{(a, 2),(a, 3),(b, 5),(-b, 7)\} \). The inputs are \(a, b,\) and \(-b\), and the outputs are \(2, 3, 5,\) and \(7\).

Step 3: Check for Multiple Outputs for a Single Input

Look at the ordered pairs in \(S\): both \((a, 2)\) and \((a, 3)\) have the same input \(a\) but different outputs \(2\) and \(3\). This violates the definition of a function where each input should map to only one output.

Step 4: Conclusion

Since the input \(a\) maps to two different outputs, \(S\) does not satisfy the condition required to be a function.

Key Concepts

Ordered Pairs

In algebra, ordered pairs are written as \((x, y)\), where \(x\) represents the input, and \(y\) represents the output. These pairs help us to see the relationship between inputs and outputs in a function clearly.
Understanding ordered pairs is crucial because it allows us to identify how elements are paired with each other. For example, in the set \( S = \{(a, 2), (a, 3), (b, 5), (-b, 7)\} \), the first element of each pair is considered the input, and the second element is the output.

• The first element, or input, can be any value such as numbers, symbols, or letters.
• The second element, or output, reflects the result connected to that specific input.

Ordered pairs are foundational in defining relationships between datasets, especially in functions.

Mapping Inputs to Outputs

Mapping is the process of connecting inputs to outputs. In mathematics, understanding how each input corresponds to an output is critical in determining whether a relationship constitutes a function.
Imagine a vending machine. You press a button (input), and a snack drops down (output). In a perfect world, each button should correspond to exactly one snack. Similarly, in math, each input should point to exactly one output.

• If two inputs lead to the same output, it is still consistent with the mapping rules.
• However, a single input cannot lead to multiple outputs while being considered a function.

Examining our set \(S = \{(a, 2), (a, 3), (b, 5), (-b, 7)\}\), notice that the input \(a\) leads to both outputs 2 and 3. This inconsistency prevents \(S\) from being a function.

Definition of a Function

In algebra, a function is a special type of relationship between inputs and outputs. For a set of ordered pairs to qualify as a function, each input must map to exactly one output.
Here are some important points about functions:

• A function can have different inputs leading to the same output.
• However, no single input can correspond to more than one distinct output.
• When plotting on a graph, a vertical line should intersect a curve at most once if it represents a function. This is known as the vertical line test.

With our example set \(S = \{(a, 2), (a, 3), (b, 5), (-b, 7)\}\), we find that since the input \(a\) maps to both 2 and 3, this set violates the rule that defines a function. Consequently, \(S\) is not a function.