Final answer:
A function in mathematics describes a relationship where each input (x) is paired with exactly one output (y). In a linear function, this is formulated with a linear equation such as y = mx + b. A function requires that every x has a corresponding y, but not necessarily the other way around.
Step-by-step explanation:
A function in mathematics describes a specific type of relationship between sets of numbers where each input value (or x) is associated with one, and only one, output value (or y). This means that in a function for every value of x, there is a single corresponding value of y. A linear function, which is often used in economic models, can be represented algebraically with a linear equation of the form y = mx + b, where m is the slope and b is the y-intercept. Graphically, this relationship can be expressed by plotting data pairs on a coordinate grid, forming a straight line, if the relationship is indeed linear.
When considering the options provided in the question:
- a. Each x is associated with multiple y values. This is not true for a function, as functions require that each x has only one corresponding y value.
- b. Each y is associated with multiple x values. This can be true for functions, as nothing in the definition of a function prohibits this.
- c. One-to-one correspondence between x and y. This is true for some functions, specifically, one-to-one functions, but not all functions must exhibit this characteristic.
- d. Some x values have no corresponding y values. This is not correct, as a function must have a corresponding y value for each element in its domain.