Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Let \(S\) be the function that assigns to each living person a social security number. Is \(S 1\) -to-1? Is it invertible? (b) Let \(C\) be the counting function that allows a collection of 30 people to be put in six groups of five people each by "counting off" 1 to 6 . Is \(C 1\) -to-1? Is it invertible? (c) Let \(A\) be the altitude function that assigns to each point in the White Mountains its altitude. Is \(A 1\) -to-1?

Step-by-Step Solution

Answer

(a) The function \(S\) is one-to-one but not invertible. (b) The function \(C\) is neither one-to-one nor invertible. (c) The function \(A\) is not one-to-one, and we lack enough information to determine if it's invertible.

Step 1: (a) Analyzing the Social Security Number Function

The social security number, denoted as function \(S\), is assigned to each living person. By definition, each person has a unique social security number and no two people share a number, which makes this function one-to-one. However, it is not invertible because there are possible social security numbers that do not correspond to a living person (i.e., the function is not onto).

Step 2: (b) Analyzing the Counting Function

The counting function \(C\) allows a collection of 30 people to be put into six groups of five each. Since multiple people can be assigned the same group number, this function is not one-to-one. Moreover, it is not invertible because not every possible grouping corresponds to a specific person.

Step 3: (c) Analyzing the Altitude Function

The altitude function \(A\) assigns to each point in the White Mountains its altitude. It is not one-to-one because multiple points can have the same altitude (e.g., points on a flat horizontal plane). However, it's not possible to say whether this function is invertible without knowing more about the specific potential range of altitudes.

Key Concepts

One-to-One Function

A function is considered one-to-one (or injective) if every element in the domain maps to a unique element in the codomain. This means no two different elements in the domain can map to the same element in the codomain. It can be understood as the rule of having a unique output for every input.
For example, consider the function assigning social security numbers to people. This assignment is one-to-one because each person has a unique social security number and no two people can share the same number.

• Each input has a different output.
• No two inputs will map to the same output.

Understanding whether a function is one-to-one helps in determining if it’s possible to "track back" from the output to the exact input, which plays a crucial role in definability of inversion.

Invertible Function

A function is invertible (or bijective) if it is both one-to-one and onto. An invertible function allows us to reverse the mapping between the domain and codomain, so each element can directly trace back to one input.
We saw that while the social security number function is one-to-one, it's not invertible because it is not onto. There are numbers not matched to any person, meaning not every element in the codomain is an output.

• A function must be injective (one-to-one) and surjective (onto) to be invertible.
• Being invertible implies that each element has a unique pre-image.

Invertibility aids in reconstructing the input from the output, ensuring a precise relationship between input and output.

Counting Function

Counting functions are often used to categorize objects or people into groups. The counting function mentioned divides 30 people into six groups of five by "counting off" numbers 1 to 6. This function is not one-to-one since multiple people can be allocated the same group number.
Instead, the same number (such as 1) can be assigned to the first person in each group repeatedly, showing repetition in outputs. This lack of uniqueness in mapping means it's also not invertible.

• Not every possible output number signifies a single group.
• Each number can map back to different inputs.

Functions like this simplify organizing items into manageable slots but lack the properties required for invertibility.

Altitude Function

The altitude function assigns a height to each point in a geographical region like the White Mountains.
Given that many points could share the same height, this function isn't one-to-one. For instance, any flat surface or ridge will lead to different points having identical altitudes.

• Multiple inputs result in the same altitude output.
• Not unique, therefore not injective.

The challenge in such situations is defining "invertibility," as the mapping is not onto and does not meet all necessary criteria without additional information.