Let's go through each statement to determine which ones are true for functions:

1. All functions have a dependent variable.

   • True: In a function, one variable (the dependent variable) depends on the other (the independent variable).

2. All functions have an independent variable.

   • True: A function requires an independent variable, which is the input for the function.

3. The range of a function includes its domain.

   • False: The domain is the set of all possible input values, while the range is the set of all possible output values. These are separate concepts; the range doesn't include the domain.

4. A vertical line is an example of a functional relationship.

   • False: A vertical line fails the vertical line test for functions because it associates one input with multiple outputs.

5. A horizontal line is an example of a functional relationship.

   • True: A horizontal line passes the vertical line test, so it is a function.

6. Each output value of a function can correspond to only one input value.

   • False: This describes a one-to-one function, but not all functions are one-to-one. Some functions allow multiple inputs to have the same output.

Summary of answers:

• True: Statements 1, 2, 5
• False: Statements 3, 4, 6

Would you like more details on any of these points?

Here are five related questions to expand your understanding:

1. What is the vertical line test, and how does it determine if a relation is a function?
2. Can a function have the same output for different input values?
3. How is the domain of a function different from its range?
4. What is an example of a one-to-one function?
5. How does a horizontal line function differ from a non-horizontal linear function?

Tip: Remember that the domain of a function is the set of all input values, and the range is the set of all output values.