Problem 46 - Algebra 2

Question

A function consists of the pairs \((2,3),(x, 4)\) and \((5,6) .\) What values, if any, may \(x\) not assume?

Step-by-Step Solution

Answer

The value of \(x\) may not be 2 or 5.

Step 1: Identify duplicate output

In a function, a particular input should point out to a single unique output. In the question provided, there are three inputs: 2, x and 5 which give the outputs 3, 4 and 6 respectively. If x was either 2 or 5, there would be two pairs with the same input but different outputs which violates the definition of a function. Therefore, we need to ensure that the x values does not equal 2 or 5.

Step 2: Conclusion

From the above step, it is determined that the value of \(x\) should not be the same as the other inputs (2 or 5) in order to preserve the definition of the function.

Key Concepts

Understanding Input-Output Pairs

In mathematics, a function is essentially a rule that takes each input and relates it to exactly one output. This relationship is often expressed as an ordered pair, such as

• (input, output)

In the given exercise, you have pairs

• (2, 3), (x, 4), and (5, 6).

These are examples of input-output pairs.
An important thing to note is that each pair represents a mapping from an input to a specific output. For example, the input "2" explicitly maps to the output "3."
If your input changes, your output will also likely change, like an interchangeable puzzle where each piece has its definitive match.

Ensuring a Unique Output

A core property of functions is that they must assign a unique output to each input. One input should never map to multiple outputs. For example, in our exercise:

• The input "2" maps to the output "3,"
• "x" maps to "4,"
• and "5" maps to "6."

You can imagine this like a vending machine where you insert a coin, and it always returns your choice of snack.
The situation where an input has multiple outputs would violate the essence of a function. A vending machine that gives both chips and candy for the same coin would be pretty confusing, just like an input with two different outputs in math!
In this exercise, making sure that "x" doesn't duplicate an existing input (like "2" or "5"), keeps the rule of unique outputs intact.

Recognizing Domain Restrictions

When dealing with functions, it's essential to understand domain restrictions. The domain is the set of all possible inputs that a function can accept. However, sometimes certain inputs are off-limits because they would violate core properties or rules of the function.
In our exercise, we've established that the function should not have two input values resulting in different outputs. Thus, the values of "x" that would equal "2" or "5" are restricted from the domain.

• If "x" were "2," that would imply two distinct outputs ("3" and "4") for the same input.
• Similarly, if "x" were "5," it'd result in outputs "4" and "6."

Such duplication doesn't fit within the definition of a function. Therefore, one must respect the domain restriction to maintain a legitimate function. This ensures not only the validity of the function but also maintains its consistency.