Describe how a mathematical function is created. Be sure to include a discussion of dependent and independent variables, and the two sets called domain and range.
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An independent variable is the potential stimulus or cause, usually directly manipulated by the experimenter, so it could also be called a manipulative variable.
A dependent variable is the response or measure of results.
A mathematical function is a relationship between two sets of variables, typically referred to as the dependent and independent variables. The function defines how the value of the dependent variable is determined by the values of the independent variable.
The independent variable is the input to the function. It represents the values that you choose or are given. For example, if you are looking at the relationship between the distance traveled and the time taken during a car ride, time would be the independent variable. You can choose different points in time and measure the distance traveled at each of those points.
The dependent variable is the output of the function. It represents the values that are determined by the independent variable. In the car ride example, distance traveled would be the dependent variable. The distance traveled depends on the time taken for the car ride.
The set of all possible values of the independent variable is called the domain. It defines the input space of the function. In our example, the domain would consist of all possible times that the car ride could take.
The set of all possible values of the dependent variable is called the range. It defines the output space of the function. In our example, the range would consist of all possible distances that could be traveled during the car ride.
Creating a mathematical function involves establishing a relationship between the independent and dependent variables. This relationship is typically expressed through an equation or a rule. For example, if we say that the distance traveled is equal to the speed of the car multiplied by the time taken, we can write the function as:
Distance = Speed * Time
This equation represents the relationship between the independent variable (time) and the dependent variable (distance). By plugging in different values of time, we can determine the corresponding distances traveled. This equation is an example of a linear function.
Functions can take different forms, such as polynomial, exponential, trigonometric, or logarithmic. They can be defined using different mathematical operations and rules. The process of creating a function involves understanding the relationship between the variables, identifying the equation that captures this relationship, and specifying the domain and range of the function.