California Assessment of Student Performance and Progress (CAASPP) Math Practice Exam

What defines a function in mathematics?

1. A relation with multiple outputs for each input

2. A relation where each input value has exactly one output value

3. A relation that includes non-numeric values

4. A relation with no inputs

The correct answer is: A relation where each input value has exactly one output value

A function in mathematics is defined as a relation where each input value corresponds to exactly one output value. This definition ensures that for every element in the domain (the set of input values), there is a unique element in the range (the set of output values). This unique mapping is what distinguishes functions from other types of relations, where an input might yield multiple outputs. For example, if we consider the input of a number like 3 in a function that squares its input, such as f(x) = x², the output will always be a single value, 9, when the input is 3. This characteristic of having a single output for each input is fundamental in various areas of mathematics and is essential for graphing and function analysis. The other options do not satisfy the definition of a function: one suggests that a relation can have multiple outputs for a single input, which contradicts the essence of a function. Another option mentions non-numeric values, which does not pertain to the core definition of a function itself. Finally, a relation with no inputs cannot constitute a function because a function requires a set of inputs to map to its outputs. Thus, the defining trait is encapsulated in the idea of having one and only one output