Problem 111 - Algebra for College Students

Question

Consider \(y=x+2\) and \(y>x+2 .\) Explain why one of these relations is a function and the other is not.

Step-by-Step Solution

Answer

The relation \( y = x + 2 \) is a function because it assigns exactly one \( y \)-value for each \( x \)-value. The relation \( y > x + 2 \) is not a function because there are multiple \( y \)-values for each \( x \)-value.

Step 1: Identifying the First Relation

The equation given is \( y = x + 2 \). To determine if this is a function, recall that a function maps each element in the domain to exactly one element in the range.

Step 2: Checking the Function Criteria for the First Relation

For every value of \( x \), there is exactly one corresponding value of \( y \) given by \( y = x + 2 \). Thus, each input \( x \) leads to one unique output \( y \). Therefore, \( y = x + 2 \) is a function.

Step 3: Identifying the Second Relation

The inequality given is \( y > x + 2 \). This states that \( y \) can take any value that is greater than \( x + 2 \).

Step 4: Checking the Function Criteria for the Second Relation

For each \( x \) value, there are multiple possible values for \( y \) (all values greater than \( x + 2 \)). Since a function must assign exactly one output to each input, \( y > x + 2 \) is not a function.

Key Concepts

Function Definition

Understanding functions is crucial in algebra. A function creates a specific mapping of each input (or domain value) to exactly one output (or range value). This means for every input value there is one and only one corresponding output value. For example, in the function given by the equation \( y = x + 2 \), every value of \( x \) generates a unique value of \( y \). This uniqueness is the defining attribute of a function: one input, one output. This concept ensures predictability in algebraic operations, making functions fundamental in mathematical studies.

Domain and Range

When discussing functions, it's important to understand what the domain and range are. The domain refers to all possible input values for the function, while the range details all possible output values.

In \( y = x + 2 \), the domain consists of all real numbers because you can substitute any real number for \( x \). As a result, the output or range is also all real numbers. This is because adding 2 to any real number still results in a real number.

On the other hand, let's analyze the inequality \( y > x + 2 \). Here, the domain is still all real numbers as any number can be substituted for \( x \). However, the range is all real numbers greater than \( x + 2 \). Hence, the inequality does not provide a unique output for each input, violating the function criterion.

Inequality Relations

Inequalities like \( y > x + 2 \) describe a set of possible solutions rather than a specific one-to-one mapping. Inequalities are less restrictive than functions because they allow multiple outputs for a single input value.

Take \( y > x + 2 \) for instance: when \( x = 1 \), any value of \( y \) that is greater than 3 (like 4, 5, or 6) will satisfy the inequality. This collection of possible \( y \) values results in a range that isn't uniquely determined by any single value of \( x \).

Such trait contrasts with functions, which demand that each \( x \) value maps exactly one corresponding \( y \) value. Recognizing this difference helps to understand why \( y = x + 2 \) is a function while \( y > x + 2 \) is not.