Assessment and Learning in Knowledge Spaces (ALEKS) Practice Exam

Define the term "function" in mathematical terms.

1. A set of ordered pairs

2. A linear equation

3. A relation that assigns exactly one output for each input

4. A static value

The correct answer is: A relation that assigns exactly one output for each input

A function in mathematics is defined as a relation that uniquely associates each input with exactly one output. This means that for every element in the domain (the set of inputs), there is a corresponding element in the range (the set of outputs), and no input can be linked to more than one output. This unique association is a crucial aspect of functions, distinguishing them from more general relations, where an input can be tied to multiple outputs. The idea of inputs and outputs reflects the notion of a process or rule that transforms the input into an output. For example, if we consider the function \( f(x) = x + 2 \), for every specific value of \( x \), this function will provide exactly one corresponding output by adding 2. In contrast, a set of ordered pairs alone does not guarantee that each first element is linked to only one second element, which is necessary to define a function. A linear equation can represent a function but does not encompass all types of functions, as functions can also be nonlinear. Additionally, a static value does not involve a relationship between inputs and outputs; it simply remains constant and does not vary. Thus, the definition that captures the unique input-output relationship accurately characterizes what a function is in mathematics.