Question

Which of the following are functions? If not, why not? (a) \(X\) is the set of students in the discrete mathematics class. For \(x \in X,\) define \(g(x)\) to be the youngest cousin of \(x\). (b) \(X\) is the set of senators serving in \(1998 .\) For \(x \in X,\) define \(g(x)\) to be the number of terms a senator has held. (c) For \(x \in \mathbb{R},\) define \(g(x)=|x /| x||\).

Short answer

(b) is a function; (a) and (c) are not.

Step-by-step solution

Understanding function criteria

A relation is a function if every input corresponds to exactly one output. We need to examine whether this condition is met for each scenario.

Analyze scenario (a)

In scenario (a), each student is mapped to 'the youngest cousin'. A student could have more than one cousin or no cousins, which violates the requirement that each input must have exactly one output. Thus, it is not a function.

Analyze scenario (b)

In scenario (b), the function maps each senator to the number of terms they have served. Since each senator serves a defined, singular number of terms, this condition is met, making it a function.

Analyze scenario (c)

In scenario (c), the function is described by the expression \(g(x) = |x / |x||\). The expression simplifies to \(g(x) = 1\) if \(x eq 0\) and is undefined at \(x = 0\). Functions need to be well-defined for every input in their domain, hence this is not a function on \(\mathbb{R}\).

Key concepts

Relation vs Function

Understanding the difference between relations and functions is crucial in discrete mathematics. A relation is simply a set of ordered pairs. In contrast, a function is a special kind of relation where every element in the domain (or the input set) is associated with exactly one element in the codomain (or the output set). This means that each input has a unique output.

In the context of discrete mathematics, consider a scenario where we have a set of students and their respective cousins. If we attempt to map each student to their cousin, but a student has multiple cousins, we are no longer dealing with a function, but instead, a relation. This is because the same student could potentially be paired with multiple cousins, violating the "exactly one" output criterion of functions.

To summarize:

• A relation can have multiple outputs for a single input.
• A function can have only one output for a single input.
• This uniqueness criterion separates functions from general relations.

Function Criteria

To determine if a relation is indeed a function, several criteria must be met. The most fundamental rule is that each input must correspond to exactly one output. Let's break down this concept further.

**Criteria for Functions:**
1. **Unique Mapping:** Each element in the domain must map to one and only one element in the codomain. For example, if a function maps from students to their ages, each student should have one distinct age associated with them. 2. **Defined Outputs for All Inputs:** Each input value from the domain must have an image in the codomain. If there's any element in the domain without a mapping, it's not a function.

Consider the scenario where senators are mapped to the number of terms they've served. Since each senator has served a specific number of terms, the mapping meets the criteria of unique mapping. Thus, this relation is a function.

To evaluate if an expression is a function, it should be analyzed across its entire domain. If for any input, the output is undefined (as is the case when dividing by zero), the relation would not qualify as a function.

Mathematical Analysis

Mathematical analysis involves a keen observation of expressions and mappings to verify if they satisfy the conditions for being considered functions. In the examination of functions, one often encounters limitations in their definition or scope.

For instance, the expression \(g(x) = |x / |x||\) is a classic example where mathematical analysis helps illuminate if the given expression holds as a function across its domain. This expression simplifies to \(g(x) = 1\) when \(x eq 0\). However, at \(x = 0\), the function becomes undefined because you cannot divide by zero. Such an undefined point in the domain indicates that this expression is not well-defined for all real numbers, disqualifying it from being a function over \(\mathbb{R}\).

**Key Points in Analysis:**

• Check if every input caresses a defined output.
• Identify any points of discontinuity or undefined behavior in expressions.
• Ensure completeness in function definition over the domain of interest.

Through thorough mathematical analysis, students can understand why certain mappings do or do not meet the stringent demands of functions.