Question

the following statements claim to describe properties that all functions possess. for each statement, use the definition of function to determine whether it is true or false. (a) if elements a and b in the domain of a function f are equal, then their images f(a) and f(b) are equal in the co-domain of f. true false (b) if elements f(a) and f(b) in the co-domain of a function f are equal, then their preimages a and b in the domain of f are equal. true false (c) a function can have the same output for more than one input. true false (b) a function can have the same input for more than one output. true false

Answer 1

Evaluating the properties of functions: statement (a) is true as equal elements in the domain lead to equal outputs in the co-domain, statement (b) is false as equal outputs do not imply equal inputs, statement (c) is true as functions can have multiple inputs producing the same output, and statement (d) is false as a function's input can only produce one output.

Step-by-step explanation:

In mathematics, functions have specific properties related to their domains and co-domains. A function is defined as a relation between a set of inputs (domain) and a set of possible outputs (co-domain), where each input is related to exactly one output. Let's analyze the given statements based on this definition:

1. (a) True: If two elements a and b in the domain of a function f are equal (a = b), then their images f(a) and f(b) are also equal. This is by the definition of a function, as each element of the domain is associated with a single element of the co-domain.
2. (b) False: If elements f(a) and f(b) in the co-domain of a function f are equal, it does not necessarily mean that their preimages a and b in the domain are equal. Functions can have different inputs that produce the same output, known as a many-to-one relation. The correct statement would be: A function can have different elements in the domain that have the same image in the co-domain.
3. (c) True: A function can indeed have the same output for more than one input. This is the essence of a many-to-one relation.
4. (d) False: A function cannot have more than one output for a given input. This would violate the definition of a function. The correct statement would be: Each element of a function's domain is associated with exactly one element of its co-domain.