Question

Which of the following statements are true about functions and relations?

1.All functions are relations.
2.All relations are functions.
3.A function may or may not be a relation.
4.The vertical line test will not work for piece-wise defined relations.

Answer 1

All functions are relations, but not all relations are functions. A function is always a relation with a unique output for each input. The vertical line test can be used for piece-wise defined functions to determine whether they are functions.

Step-by-step explanation:

In the context of mathematical functions used in economic models or otherwise, here is the clarification of the statements about functions and relations:

1. All functions are relations. This statement is true. Every function defines a relationship between sets of information such that each input is related to exactly one output.
2. All relations are functions. This statement is false. A correct statement would be: A relation may or may not be a function, depending on whether every input has a unique output.
3. A function may or may not be a relation. This statement is false. A correct statement would be: A function is always a relation, but with the additional constraint that each input has one and only one output.
4. The vertical line test will not work for piece-wise defined relations. This statement is false. A correct statement would be: The vertical line test can work for piece-wise defined functions to determine if they are indeed functions, by checking if a vertical line intersects the graph more than once.