Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Which of the following functions are invertible? (a) The function that assigns to each current senator the state he or she represents. (b) The function \(T(t)\) that gives the temperature in Moab at time \(t\). (c) The function \(C(d)\), whose domain is the set of all performances of Broadway's \(A\) Chorus Line, and whose output is the cumulative number of people who have seen this show on Broadway. (d) The function \(L(d)\), whose domain is the set of all performances of Broadway's The Lion King, and whose output is the number of people seeing this Broadway show on the designated date.

Step-by-Step Solution

Answer

Functions (c) and (d) are invertible while functions (a) and (b) are not.

Step 1: Analysis of Function (a)

The function assigns to each current senator the state he or she represents. There might be several senators representing the same state and thus, the function isn't injective. Therefore, this function is not invertible.

Step 2: Analysis of Function (b)

The function \(T(t)\) that gives the temperature in Moab at time \(t\) can be different at different points of time t, however, the same temperature could occur at more than one point of time. Therefore, this function isn't injective - hence, it is not invertible.

Step 3: Analysis of Function (c)

The function \(C(d)\) computes the cumulative number of people who have seen the show starting from the first performance. As the number of people is cumulative, the number strictly increases (or stays the same) after each performance. Hence, every performance will be associated with a unique number of cumulative audiences, which makes the function injective. Therefore, it is invertible.

Step 4: Analysis of Function (d)

The function \(L(d)\) gives out the number of people seeing this Broadway show on the designated date. It seems reasonable to assume that the audience size varies from day to day - hence, every performance could be associated with a unique number of audience. This makes the function injective, and therefore, it is invertible.

Key Concepts

Injective Function

An injective function, also known as a one-to-one function, is a type of function where each element in the domain is mapped to a unique element in the codomain. This means if we have two different elements in the domain, they will map to two different elements in the codomain.
For a function to be injective, for any two elements \(x_1\) and \(x_2\) in the domain, if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\). In simpler words, no horizontal line on the graph of the function should intersect it at more than one point.

• Injective function ensures that no two distinct inputs give the same output.
• Being injective is a necessary condition for a function to be invertible.
• Understanding injective functions helps analyze and determine the invertibility of various functions.

By understanding injective functions, determining if a function can be inverted becomes much clearer. If a function lacks this property, it cannot be inverted since it doesn't produce a unique output, critical for defining a reverse mapping.

Function Analysis

Function analysis involves examining a function's behavior and characteristics to determine its properties. This helps in identifying if a function is injective, surjective, or bijective.
Analyzing a function involves looking at several key aspects:

• Domain and Codomain: Know what inputs the function can take and what possible outputs it can give.
• One-to-one Correspondence: Determine if the function is injective, surjective, or bijective.
• Continuity and Discreteness: Check if the function is continuous or discrete, which impacts its invertibility.

Function analysis is the foundation for understanding complex mathematical concepts, as it functions as the groundwork for discussing invertibility and other characteristics.

Mathematical Inverses

A function's inverse is essentially the reverse process of the original function. If \(f\) is a function mapping elements from \(X\) to \(Y\), then the inverse of \(f\), denoted as \(f^{-1}\), maps elements back from \(Y\) to \(X\).
To determine if a function has an inverse, it first needs to be injective (one-to-one) and surjective (onto).

• Injectivity: Ensures each element in the domain maps to a unique element in the codomain.
• Surjectivity: Every element in the codomain is mapped by an element from the domain.
• Bijective: A function that is both injective and surjective, making it fully invertible.

If both conditions are satisfied, the inverse of the function can be defined that reverses the effect of the function. Calculating a mathematical inverse is vital in solving equations and understanding deeper relationships between variables. When a function has an inverse, it’s often an indicator of its richness and utility in various mathematical applications.