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Chapter 1: Problem 1

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

Short Answer

Expert verified
Short Answer: A function is a relationship between an independent variable (input) and a dependent variable (output), where the dependent variable depends on the value of the independent variable. The domain consists of all possible independent variable values for which the function is defined, while the range consists of all possible dependent variable values that can be obtained by inputting the domain values into the function. For example, in the linear function y = 2x + 3, x is the independent variable, and y is the dependent variable. The domain and range are both all real numbers, and as the value of x changes, the value of y will also change according to the equation, demonstrating the relationship between the two variables.

Step by step solution

01

Define the terms

- Domain: The domain of a function consists of all possible values of the independent variable for which the function is defined. - Range: The range of a function consists of all possible values of the dependent variable that can be obtained by plugging the values of the independent variable from the domain into the function. - Independent variable: The independent variable is the input of a function, which can be manipulated without being influenced by any other variables. It is usually represented by x. - Dependent variable: The dependent variable is the output of a function, which depends on the value of the independent variable. It is usually represented by y.
02

Provide an example of a function

Let's consider a linear function, for example: y = 2x + 3 In this function, x is the independent variable and y is the dependent variable.
03

Identify the domain and range

For linear functions, the domain and range are usually all real numbers, unless there are specific restrictions given. For the given function, there are no restrictions, so the domain and range are as follows: - Domain: The domain of the function is all real numbers, which can be represented as (-鈭�, +鈭�) in interval notation. - Range: Since the domain is all real numbers and there are no restrictions on the values that y can take, the range is also all real numbers, which can be represented as (-鈭�, +鈭�) in interval notation.
04

Explain how the function relates the independent and dependent variables

In the example function y = 2x + 3, the dependent variable (y) is related to the independent variable (x) through the equation. As the value of x changes, the value of y will also change according to the equation. The domain represents all possible values of the independent variable (x) that can be plugged into the function, while the range represents all possible output values (y) that we can get after inputting the values from the domain. In conclusion, a function relates one variable (the dependent variable) to another variable (the independent variable) through an equation. By defining its domain and range, we can understand the relationship between these variables, as the function's output (dependent variable) depends on the input value (independent variable).

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