Question

Explain the meaning of the terms 'domain' and 'range' when applied to functions.

Short answer

Answer: The domain of a function refers to the set of all possible input values (independent variables) for which the function is defined. The range of a function refers to the set of all possible output values (dependent variables) that can be obtained by inputting values from the domain into the function. In other words, the domain is associated with the input values, and the range is associated with the output values that the function can produce.

Step-by-step solution

Domain

The domain of a function is the set of all possible input values (also called independent variables) for which the function is defined. In other words, it is the set of all values that can be plugged into the function to get a valid output or result. For example, if we have a function f(x) = 1/x, the domain would be all real numbers except 0, as dividing by 0 is undefined.

Range

The range of a function is the set of all possible output values (also called dependent variables) that can be obtained by inputting values from the domain into the function. In other words, it is the set of all results that can be produced by the function. For example, if we have a function f(x) = x^2, the range would be all non-negative real numbers, as when squaring any real number, the result will always be greater than or equal to 0.

Summary

Domain and range are important concepts in understanding how functions behave and what values they can accept and produce. Remember, the domain of a function refers to the set of all possible input values, while the range refers to the set of all possible output values obtained from the input values in the domain.

Key concepts

Domain

When talking about the domain of a function, think of it as the collection of all possible inputs that can go into the function. These inputs are often referred to as independent variables. For a function to make sense, each value you put in must be valid. Take, for instance, the function \( f(x) = \frac{1}{x} \). In this case, the domain consists of all real numbers, but we must exclude 0. This is because division by 0 is impossible, rendering it outside the function's domain. So, understanding a function's domain helps in identifying which inputs are permissible.

Range

The range of a function is like the flip side of the domain. Rather than focusing on the inputs, the range is about the outputs. It shows all the possible results you can get after plugging in values from the domain. These outputs are also known as dependent variables because they rely on the input values. For example, consider the function \( f(x) = x^2 \). Here, no matter what real number you substitute for \( x \), the result will always be a non-negative number. Therefore, the range includes all non-negative real numbers, starting from 0 onwards. Recognizing a function's range provides insight into its behavior and the outputs you can expect.

Independent Variables

Independent variables are like the control knobs of a function. They determine what value you put into the function. Whenever you see a function written like \( f(x) \), the \( x \) represents the independent variable. It’s independent because you are free to choose any value for \( x \) within the function's domain. For instance, in a function describing the height of a plant over time, time could be the independent variable. You decide what point in time to examine, and the function tells you about the plant's height then. Grasping the role of independent variables aids in controlling the inputs to your function.

Dependent Variables

Dependent variables come into play once you've chosen your independent variable. They’re called "dependent" because their value depends on the choice of the independent variable. In terms of a function, the output or result is the dependent variable. For instance, if a function describes how far a car travels based on time, the distance traveled is the dependent variable. The calculation of distance depends on the input value of time. Therefore, dependent variables provide essential information about the outcomes and results of the function based on the input selections.