Final answer:
The statements which are true : All functions have a dependent variable, All functions have an independent variable and A horizontal line is an example of a functional relationship.
Step-by-step explanation:
The question revolves around understanding the nature of mathematical functions and their properties. In mathematics, a function is a relationship between two sets that associates each element of the first set (the domain) with a unique element of the second set (the range). Let's address the given statements:
- All functions have a dependent variable. True: In a function, the output (dependent variable) depends on the input (independent variable).
- All functions have an independent variable. True: The input of a function is known as the independent variable.
- The range of a function includes its domain. False: The domain is the set of all possible inputs, while the range is the set of all possible outputs. The range cannot include the domain as they are distinct aspects of a function.
- A vertical line is an example of a functional relationship. False: A vertical line fails the vertical line test, which means that for some input (x-value), there is more than one output (y-value). This violates the definition of a function.
- A horizontal line is an example of a functional relationship. True: A horizontal line represents a function since for every input, there is exactly one output.
- Each output value of a function can correspond to only one input value. False: This statement is the converse of the definition of a function. A function requires that each input corresponds to exactly one output, but multiple inputs can produce the same output.