Let P be the set of all people living in the United States....

Let $P$ be the set of all people living in the United States. Give examples of each of the following functions and state their ranges: (a) the identity function $f: P \rightarrow P$ (b) two different constant functions $g: P \rightarrow P$ (c) a non-constant, non-identity function $g: P \rightarrow P$ (d) a constant function $h: P \rightarrow \mathbb{R}$ (e) a non-constant function $h: P \rightarrow \mathbb{R}$

Let $P$ be the set of all people living in the United States. Give examples of each of the following functions and state their ranges:
(a) the identity function $f: P \rightarrow P$
(b) two different constant functions $g: P \rightarrow P$
(c) a non-constant, non-identity function $g: P \rightarrow P$
(d) a constant function $h: P \rightarrow \mathbb{R}$
(e) a non-constant function $h: P \rightarrow \mathbb{R}$

Key Concepts

Non-Constant Function

A non-constant function is a function that does not map every element of the domain to a single, fixed output. Instead, the function produces at least two distinct outputs for different inputs, demonstrating variation and dependence on the input.

Constant Function

A constant function is one in which every element of the domain is mapped to the same fixed value in the codomain. This means that regardless of the input, the output remains the same, which is important when demonstrating functions that do not vary with different inputs.

Range of a Function

The range of a function is the set of all output values that the function actually produces from its domain. Knowing a function's range helps in understanding which elements of the codomain are being ‘hit’ by the function, differentiating it from the full codomain.

Domain and Codomain

The domain is the set of all possible input values for a function, while the codomain is the set into which all outputs fall. In problems involving functions, specifying these sets is crucial for understanding the behavior and possible outputs of the function.

Identity Function

An identity function is a specific type of function for which every element in the domain maps to itself. It acts as a neutral element in function composition, meaning that applying the identity function to an element returns that same element without any alteration.

Function

A function is a rule that assigns each input element from a set called the domain a unique output element in a set called the codomain. Functions are central in mathematics as they establish clear mappings between elements of different sets.

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