Problem 12 - Precalculus

Question

Determine whether or not the relation represents \(y\) as a function of \(x .\) Find the domain and range of those relations which are functions. $$ \left\\{\left(x^{2}, x\right) \mid x \text { is a real number }\right\\} $$

Step-by-Step Solution

Answer

The relation is not a function because \(x^2\) can map to both \(x\) and \(-x\).

Step 1: Understanding the Relation

The given relation is \( \{(x^2, x) \mid x \text{ is a real number}\} \). This means for each real number \(x\), it is paired with \(x^2\).

Step 2: Determining if it's a Function

For a relation to be a function, each input \(x^2\) must map to exactly one output \(x\). However, the same \(x^2\) can correspond to two different \(x\) values (both \(x\) and \(-x\)). Therefore, this is NOT a function.

Step 3: Final Step: Conclusion

Since the relation is not a function, we do not need to determine the domain and range. The original question asks this only if the relation is a function. Thus, the exercise is complete.

Key Concepts

Domain and Range

In mathematics, the concepts of domain and range are essential when studying functions and relations.
The domain of a function or relation is the set of all possible input values. These inputs are often denoted by the variable \( x \), and they must satisfy the conditions of the function or relation.

The range, on the other hand, is the set of all possible output values, determined by plugging the domain values into a function or relation. The range is often represented by the variable \( y \).

• For example, if we have a relation like \((x^2, x)\), the domain is all possible \( x \) values that we can use in the relation, which in this case, is all real numbers, since \( x \) can be any real number.
• The range for this relation would be all resulting \( x^2 \) values, which are always non-negative for real \( x \).
  

Understanding these concepts helps to describe a relation fully. Knowing the domain tells you where the relation exists, while the range tells you its outputs.

Real Numbers

Real numbers are a fundamental element in mathematics, forming the basis of most mathematical analyses. They include:

• Rational numbers, which can be expressed as fractions with integer numerators and non-zero integer denominators.
• Irrational numbers, such as \( \pi \) and \( \sqrt{2} \), which cannot be expressed exactly as fractions.

The set of real numbers also includes:

• Whole numbers (0, 1, 2, 3, ...)
• Integers (..., -2, -1, 0, 1, 2, ...)
• Positive and negative decimals

In the context of the given exercise, \( x \) represents a real number, which means it can take any value from this wide set. This allows the relation \((x^2, x)\) to include values from across the entire range of real numbers, making the analysis of such relations broad and comprehensive.

Mathematical Relations

In mathematics, relations define how two sets of information relate to one another. A relation consists of pairs of input and output values. In a relation, each input \( x \) from one set is related to one or more outputs \( y \) from another set.

In our exercise, the relation \((x^2, x)\) indicates a specific pairing where each input \( x \) results in an output \( x^2 \). The concept of a relation is essential because it sets the stage for understanding functions. If every input corresponds to exactly one output, the relation becomes a function.

• For example, \((1,1)\) and \((1,-1)\) are pairs in our given relation. This highlights that one output \( 1 \) comes from two different inputs, \( 1 \) and \(-1 \).
  

This means our relation is not a function because it violates the unique output condition. Understanding the differentiation between relations and functions is vital for deeper insights into mathematical structures.