Question

Use the terms domain, range, independent variable, and dependent variable to explain how a function relates one variable to another variable.

Short answer

Answer: In the function f(x) = 2x + 3, the independent variable is x and the dependent variable is f(x). The domain of this function is the set of all real numbers, represented as D = (-∞, ∞), and the range is also the set of all real numbers, represented as R = (-∞, ∞). The relationship between the independent and dependent variables is determined by the formula f(x) = 2x + 3, which means that for every value of x from the domain, there is a unique corresponding value of f(x) in the range.

Step-by-step solution

Define the terms

Domain: The domain of a function is the set of all possible input values (independent variables) for which the function is defined. Range: The range of a function is the set of all possible output values (dependent variables) that the function can produce. Independent Variable: An independent variable is the input variable in a function that can be freely chosen from the domain. It is the variable that is being manipulated to analyze its effect on the dependent variable. Dependent Variable: A dependent variable is the output variable in a function that depends on the value of the independent variable. It is the variable that is being studied or measured in the relationship.

Provide an example

Let's consider the function f(x) = 2x + 3, where x is the independent variable and f(x) is the dependent variable. Step 1: Identify the domain and range Here, the domain of the function is the set of all possible values of x for which the function is defined. In this case, the domain is all the real numbers, denoted by D = (-∞, ∞). The range of the function is the set of all possible values of f(x) that the function can produce. As x can be any real number, f(x) can also be any real number, so the range is also R = (-∞, ∞). Step 2: Evaluate the relationship between the independent and dependent variables In this function, the dependent variable f(x) relies on the value of the independent variable x. For every value of x chosen from the domain, there is a corresponding value of f(x) in the range which is determined by the formula f(x) = 2x + 3. For example: - If x = 1, then f(1) = 2(1) + 3 = 5 - If x = -2, then f(-2) = 2(-2) + 3 = -1 In conclusion, a function relates an independent variable to a dependent variable by defining a rule that assigns each value from the domain to a unique value in the range. The independent variable represents the input values of a function, while the dependent variable represents the output values, which are determined by the function's rule.