Final answer:
In terms of functions, it's true that all functions have a dependent and an independent variable, and a horizontal line represents a functional relationship. It's incorrect to say that the range includes the domain, that a vertical line represents a function, or that each output value corresponds to only one input.
Step-by-step explanation:
Let's address each statement in regard to functions:
- All functions have a dependent variable: This is true because in a function, each input from the domain is associated with an output in the range, and this output is called the dependent variable.
- All functions have an independent variable: This is also true. In a function, the independent variable is the input value, commonly represented as 'x', from which the dependent variable or output 'y', is derived.
- The range of a function includes its domain: This statement is false. The domain is the set of all possible input values, while the range is the set of all possible outputs. They are correlated, but one is not inclusive of the other.
- A vertical line is an example of functional relationship: This is false because a vertical line would fail the vertical line test. It suggests that a single input could be associated with multiple outputs, which is not allowed in functions.
- A horizontal line is an example of a functional relationship: This is true, as a horizontal line passes the vertical line test, meaning that for each input value there is only one output value.
- Each output value of a function can correspond to only input value: This statement is false. It's the reverse; each input value of a function should correspond to only one output value.
In summary, a function in mathematics describes a specific relationship where each input (independent variable) is associated with exactly one output (dependent variable). Economic models often use functions to define relationships between variables.