Question

Which statement about a function is true? (i) For each value of \(x\) in the domain, there corresponds one unique value of \(y\) in the range; (ii) for each value of \(y\) in the range, there corresponds one unique value of \(x\) in the domain. Explain.

Short answer

Answer: Statement (i) is true about a function because it states that "For each value of x in the domain, there corresponds one unique value of y in the range." This is in accordance with the definition of a function, where each input value (from the domain) corresponds to a unique output value (from the range).

Step-by-step solution

Analyze Statement (i)

For a given function, a value of x corresponds to a unique value of y. This is based on the definition of a function, which states that each element in the domain (input values) is mapped to exactly one element in the range (output values).

Analyze Statement (ii)

Statement (ii) claims that for each value of y in the range, there corresponds one unique value of x in the domain. This statement is not true for every function, as some functions may have multiple inputs that yield the same output.

Compare the Statements

Statement (i) is in accordance with the definition of a function, where each input value (from the domain) corresponds to a unique output value (from the range). On the other hand, statement (ii) is incorrect since it claims that each output value (from the range) corresponds to a unique input value (from the domain), which isn't necessarily the case for all functions.

Conclusion

Based on the analysis, statement (i) is the true statement about a function: "For each value of \(x\) in the domain, there corresponds one unique value of \(y\) in the range."

Key concepts

Domain and Range

When discussing functions, two important concepts come to the forefront: domain and range. The domain of a function is the complete set of possible input values, often represented by the variable \(x\). The range, on the other hand, is the complete set of possible output values, typically represented by the variable \(y\). These two sets together form the foundation of a function's operation.
Understanding domain and range helps in identifying how a function behaves. For example, a function \(f(x) = x^2\) has a domain of all real numbers because you can square any real number. However, its range consists only of non-negative real numbers because squaring any real number cannot produce a negative result.
When calling something a function, it's crucial to specify both the domain and range since they define the scope and limit of the function's application. This specification assists in predicting the behavior of functions across their defined domains and ranges.

Unique Mapping

Unique mapping is a defining characteristic of functions. In the context of functions, it ensures that each element in the domain maps precisely to one element in the range.
What does this mean? Simply put, for any input \(x\) within the function's domain, there should be one and only one corresponding output \(y\) in the range. This is crucial because it maintains the integrity and predictability of the function.
An example can clarify this concept: Let's consider the function \(f(x) = x + 5\). No matter what value \(x\) takes within its domain, it will always map to a single, unique value of \(y\). If \(x = 2\), then \(f(x) = 7\), ensuring a unique pairing of input and output.

• Every \(x\) in the domain maps to a unique \(y\) in the range.
• This concept does not necessarily apply in reverse, as multiple \(x\) values can map to the same \(y\).

Unique mappings help in maintaining a straightforward and unambiguous function definition, reinforcing the function's reliability.

Definition of a Function

At the heart of understanding functions lies their definition, which is straightforward yet foundational in mathematics. A function is essentially a rule or a relation that assigns exactly one output \(y\) for each possible input \(x\) within its domain. This one-to-one assignment to each element is what distinguishes functions from other mathematical relations.
The essence of a function’s definition is in its predictability and consistency. For every \(x\) that you input, you'll always know what \(y\) to get as output. Consider the function \(f(x) = 2x + 3\). For any \(x\), the result will be a \(y\) precisely defined by this equation, ensuring that each input will consistently yield the same output.
Functions are a structured way to relate inputs to outputs, essential across various branches of mathematics. They form the backbone of algebra, calculus, and many real-world applications, from predicting financial trends to modeling scientific phenomena. By ensuring a unique output for each input, functions make mathematical modeling both reliable and powerful.