what is the equation in this ordered pairs in gen math {(1,5),(2,5),(3,6),(4,6)}

Given the ordered pairs \(\{(1, 5), (2, 5), (3, 6), (4, 6)\}\), we want to find an equation that represents this relationship in general mathematics.

### Observing the Pairs
Let's analyze the pairs:
- The first two pairs are \((1, 5)\) and \((2, 5)\), where different \(x\) values (1 and 2) yield the same \(y\) value (5).
- The next two pairs are \((3, 6)\) and \((4, 6)\), where different \(x\) values (3 and 4) yield the same \(y\) value (6).

### Step 1: Determine the Relationship
The relationship suggests that the \(y\) value depends on the range of \(x\):
- \( y = 5 \) when \( x = 1 \) or \( x = 2 \).
- \( y = 6 \) when \( x = 3 \) or \( x = 4 \).

### Step 2: Write the Equation Piecewise
This relationship can be expressed using a piecewise function:
\[
y = \begin{cases} 
5 & \text{if } x = 1 \text{ or } x = 2 \\
6 & \text{if } x = 3 \text{ or } x = 4 
\end{cases}
\]

### Step 3: Generalize the Piecewise Function
While the above piecewise function works well for the specific pairs given, it doesn't represent a single algebraic equation for all \(x\). However, it is the correct way to describe the relation based on the provided pairs.

### Summary
The equation for the given ordered pairs is best represented as a piecewise function:
\[
y = \begin{cases} 
5 & \text{if } x = 1 \text{ or } x = 2 \\
6 & \text{if } x = 3 \text{ or } x = 4 
\end{cases}
\]

This shows the many-to-one nature of the relation, where different \(x\) values map to the same \(y\) value.

Would you like further details on how to work with piecewise functions, or do you have any other questions?

### Further Exploration:
1. What is the general definition of a piecewise function?
2. Can this piecewise function be simplified further?
3. How would the graph of this piecewise function look?
4. What is the difference between a continuous and a discontinuous piecewise function?
5. Can we represent these ordered pairs using a different type of function?
6. How would you express this relationship using a table of values?
7. What are some real-life examples of piecewise functions?
8. How can piecewise functions be used to model data in economics?

**Tip:** Piecewise functions are very useful in modeling situations where different rules apply to different intervals of the domain.

Explore piecewise functions using the example of {(1, 5), (2, 5), (3, 6), (4, 6)}. Learn how to represent multiple input-output relationships in mathematics and analyze the many-to-one nature of these ordered pairs. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation