Question

Decide whether the set of ordered pairs represents a function from \(A\) to \(B\). \(A=\\{0,1,2,3\\}\) and \(B=\\{-2,-1,0,1,2\\}\) Give reasons for your answers. \(\\{(0,1),(1,-2),(2,0),(3,2)\\}\)

Short answer

The given set of ordered pairs \{(0,1),(1,-2),(2,0),(3,2)\} does represent a function from A to B.

Step-by-step solution

Identify the ordered pairs

Look at each ordered pair in the given set. Here, the set of ordered pairs is \{(0,1),(1,-2),(2,0),(3,2)\}.

Check for repeated first elements

Analyze if there are any repeats in the first coordinates (the x-values) of the ordered pairs. In this case, the first coordinates of the ordered pairs are 0, 1, 2, and 3. None of these numbers repeat. Therefore, each number in set A is matched with a unique number in set B.

Make a conclusion

Since all elements in A are paired with exactly one element in B, and no two ordered pairs have the same first element, this set of ordered pairs can be considered a function from A to B.

Key concepts

Ordered Pairs

In algebra, an ordered pair is a fundamental concept that represents a relationship between two elements.
Each ordered pair consists of two components enclosed in parentheses, separated by a comma, such as \((a, b)\). The first element is known as the "first coordinate", and the second is the "second coordinate".

These components have a specific order; changing the order changes the pair's meaning or representation.

• The first coordinate typically represents an element from the domain (set A), which is the set of all possible inputs or "x-values."
• The second coordinate corresponds to an element from the range (set B), which is the set of all possible outputs or "y-values."

For example, in the set \(\{(0,1),(1,-2),(2,0),(3,2)\}\),
each pair is formed by an element from subset A and bounded with an element from subset B, indicating how these elements are related. The structured arrangement in ordered pairs ensures clarity in describing functions or relations.

Domain and Range

The domain and range are crucial concepts in understanding functions.
They define the set of permissible values the function can accept and produce.

The domain refers to the set of all possible inputs or starting values for a function.

• For our set \(A = \{0,1,2,3\}\), these numbers form the domain. They represent every single "x-value" or initial component in the ordered pairs.

The range, on the other hand, is the set of all possible outputs we get from the function.

• From our set \(B = \{-2,-1,0,1,2\}\), it's the collection of potential "y-values" or second components in our ordered pairs.

Understanding the domain and range helps us see how the elements from set A pair with elements from set B,
creating a network of ordered relations that can be verified as functions.

Unique Mapping

In functions, the idea of unique mapping ensures that each input from the domain can have only one corresponding output in the range.
A function from a set A to a set B must adhere to this principle of unique mapping.

Let's break down this concept using our example: each member of set A can connect with precisely one member of set B via the ordered pairs \((0, 1), (1, -2), (2, 0), (3, 2)\).

• The criterion for unique mapping requires that no two ordered pairs have the same first element.
• This ensures that a particular input from set A maps to one and only one output in set B.

Such a mapping aligns with the definition of a function. If any first element is repeated across different pairs,
it contradicts the unique mapping rule, suggesting the relationship isn’t a function. Ensuring unique mapping is vital for a set of ordered pairs to qualify as a function.