How to check if a set of points represents a function (Grade 8 guide)

📚 What is a Function?

Before we dive into checking sets of points, let's quickly recap what a function actually is. A function is a special relationship where each input (usually an 'x' value) has only one output (usually a 'y' value). Think of it like a vending machine; you press a button (input), and you get a specific snack (output). You wouldn't expect to press the same button and get two different snacks, right?

📜 History of the Function Concept

The idea of a function has evolved over centuries. Early notions were tied to geometry and curves. Mathematicians like Nicole Oresme in the 14th century started exploring the relationship between quantities. Later, mathematicians like Leibniz and Bernoulli formally defined functions, leading to the modern definition we use today.

📌 The Vertical Line Test: The Key Principle

The easiest way to check if a set of points represents a function is by using the Vertical Line Test. Imagine drawing vertical lines through the graph. If any vertical line crosses the graph more than once, then it's not a function. This is because that x-value would have multiple y-values, violating the definition of a function.

🪜 Steps to Check if a Set of Points Represents a Function:

• 📍Step 1: Plot the Points:
Plot all the given points on a graph. Make sure your x and y axes are clearly labeled.
• 📏Step 2: Draw Vertical Lines:
Imagine (or draw lightly) vertical lines through each point on the x-axis.
• ✅Step 3: Apply the Vertical Line Test:
Check if any of your vertical lines intersect the plotted points more than once. If all the vertical lines intersect at most once, the relation is a function. If any vertical line intersects more than once, then the relation is NOT a function.

➕ Real-World Examples

Let's look at some examples to see this in action.

Example 1: Function

Consider the set of points: (1, 2), (2, 4), (3, 6), (4, 8)

If you plot these points, you'll see that no vertical line crosses more than one point. This is a function.

Example 2: Not a Function

Consider the set of points: (1, 2), (2, 4), (1, 5), (3, 6)

If you plot these points, you'll see that a vertical line at x = 1 crosses both (1, 2) and (1, 5). This is not a function.

📊 Representing Points and the Vertical Line Test Mathematically

Let's say we have a set of ordered pairs $(x, y)$. A relation is a function if for every $x$ value, there is only one corresponding $y$ value. We can express the vertical line test as follows: if $(a, b)$ and $(a, c)$ are in the set, then $b$ must equal $c$. Otherwise, it's not a function.

✏️ Practice Quiz

Determine whether each set of points represents a function:

1. {(0,1), (1,2), (2,3), (3,4)}
2. {(1,1), (1,2), (2,3), (3,4)}
3. {(-1,0), (0,0), (1,0), (2,0)}
4. {(0,-1), (0,0), (0,1), (0,2)}
5. {(1,5), (2,5), (3,5), (4,5)}
6. {(-2,4), (-1,1), (0,0), (1,1)}
7. {(1,2), (2,4), (3,6), (1,8)}

Answers:

1. Function
2. Not a function
3. Function
4. Not a function
5. Function
6. Function
7. Not a function

🎉 Conclusion

Understanding functions is a cornerstone of mathematics. The Vertical Line Test provides a simple way to check if a relation represented by a set of points qualifies as a function. Keep practicing, and you'll master this concept in no time!