Problem 29 - Elementary and Intermediate Algebra

Question

Determine whether the relation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x\). See Example 2. $$ \\{(-2,7),(-1,10),(0,13),(1,16)\\} $$

Step-by-Step Solution

Answer

Yes, the relation defines \(y\) as a function of \(x\).

Step 1: Understanding Function Definition

To determine if a relation defines a function, recall that each input (each value of \(x\)) must be associated with exactly one output (one value of \(y\)). This means that in the set of ordered pairs, no \(x\) value should repeat with different \(y\) values.

Step 2: Analyzing the Given Set of Pairs

Given the set \{(-2,7),(-1,10),(0,13),(1,16)\}, examine each pair to ensure that each \(x\) value is unique. Here, the values of \(x\) are -2, -1, 0, and 1, none of which repeat.

Step 3: Conclusion

Since every \(x\) value in the set corresponds to exactly one \(y\) value, the relation does define \(y\) as a function of \(x\).

Key Concepts

Relations and Functions

In algebra, distinguishing between relations and functions is a crucial concept, especially when dealing with sets of ordered pairs. A **relation** is simply a set of ordered pairs, where each pair consists of an input value, usually denoted as \(x\), and an output value, denoted as \(y\). This format helps to establish connections between two variables.However, not every relation qualifies as a **function**. For a relation to be considered a function, each input value must correspond to exactly one output value. This means in the set of ordered pairs:

• Each \(x\) value must be unique when paired with its corresponding \(y\) value.
• No \(x\) value should have multiple \(y\) values associated with it.

If a relation meets these criteria, it is considered a function. This fundamental rule helps to maintain clarity and order when defining how two variables relate to each other.

Ordered Pairs

Ordered pairs are the foundation of understanding relations and functions. An ordered pair comprises two components:

• The first element is the \(x\)-value, which represents the input.
• The second element is the \(y\)-value, which represents the output.

Consider the ordered pair (-2, 7). Here, -2 is the \(x\)-value and 7 is the \(y\)-value. The order matters, indicating the direction of the relation from \(x\) to \(y\).By analyzing sets of ordered pairs, you can determine whether a relation is indeed a function. In the provided exercise, the set \{(-2,7),(-1,10),(0,13),(1,16)\} was examined. Each ordered pair contains a distinct \(x\) value, meaning no \(x\) is repeated, which is key the characteristic that qualifies the relation as a function. Keep in mind that the correct sequence and pairing in ordered pairs are crucial when evaluating the nature of a relation.

Unique X-Values

The uniqueness of \(x\)-values is a defining feature of functions in algebra. In a function, every \(x\) must associate with one and only one \(y\). This distinct characteristic ensures the function's reliability.Consider the set of ordered pairs \{(-2,7),(-1,10),(0,13),(1,16)\}. Here, the \(x\)-values are -2, -1, 0, and 1. Each appears only once within the set, ensuring that each \(x\)-value pairs uniquely with a \(y\)-value. This distinctiveness of \(x\)-values confirms that the set represents a function.If any \(x\) value had appeared more than once with different \(y\)-values, the set would not define a function. In summary, ensuring that \(x\)-values are unique is essential for recognizing proper functions in algebra.