Question

which of the following functions (or families of functions) are 'naturally' injective, i.e. when defined on their usual domains? a. non-constant linear functions b. power functions with an even degree c. floor and ceiling function d. logarithmic functions e. power functions with an odd degree f. polynomials g. inverse trig functions h. exponential functions i. rational functions j. trigonometric functions k. the absolute value function l. none of the above

Answer 1

The functions that are 'naturally' injective are non-constant linear functions, power functions with an odd degree, inverse trig functions, exponential functions, and rational functions.

Step-by-step explanation:

Based on the given information, the functions that are 'naturally' injective, i.e. injective when defined on their usual domains, are:

a. non-constant linear functions
b. power functions with an odd degree
g. inverse trig functions
h. exponential functions
i. rational functions

These functions have certain properties that make them one-to-one. For example, linear functions have a constant rate of change, and exponential functions have a positive base that is not equal to 1. It is important to note that not all functions in each family may be injective, but based on the information provided, these families of functions generally exhibit injective behavior.