Construct examples of the thing(s) described in the...

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A function that is defined with a formula.
(b) A function that is not defined with a formula.
(c) A formula that does not define a function.

Key Concepts

Function

A function is a specific type of relation between two sets in which every element of the first set (the domain) is paired with exactly one element of the second set (the codomain). This uniqueness condition is what distinguishes functions from more general relations.

Formula

A formula is a mathematical expression that uses symbols and operations to represent a relationship between variables. In the context of functions, a formula often serves to explicitly calculate the output for any given input.

Explicit vs Implicit Definitions of Functions

Functions can be given explicitly by means of a formula, where the rule for generating the output is directly stated, or implicitly through other means such as a graph, a table, or a descriptive rule. An explicitly defined function uses a formula, while an implicitly defined function may be identified by its properties or behaviors without a single governing expression.

Non-Function Relations and the Vertical Line Test

Not every mathematical expression or relation qualifies as a function. A relation may involve a formula that relates inputs to multiple outputs, violating the one-output-per-input rule. When represented graphically, such a relation will fail the vertical line test, where a vertical line drawn through the graph intersects it at more than one point, indicating that the expression does not define a function.

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