Range

Definition

The range of a function refers to the set of all possible output values or the set of all values that the function can attain. It represents the vertical extent or the interval of values that the function can produce as the input variable changes. The range is an important concept in the study of functions and their properties, as it provides information about the behavior and characteristics of the function.

5 Must Know Facts For Your Next Test

1. The range of a function is often denoted using the notation $\{y \mid y = f(x)\}$, where $f(x)$ represents the function.
2. The range of a function can be determined by analyzing the graph of the function, observing the highest and lowest points or values attained by the function.
3. The range of a linear function is an unbounded interval, as the function can produce any real number as an output.
4. The range of a quadratic function is a bounded interval, as the function has a maximum or minimum value.
5. The range of an absolute value function is a non-negative interval, as the absolute value of any number is always greater than or equal to zero.

Review Questions

• Explain how the range of a function is related to its domain and codomain.
  • The range of a function is a subset of the function's codomain, as it represents the set of all possible output values that the function can produce. The range is determined by the input values in the domain and the relationship between the input and output values defined by the function. While the codomain specifies the set of all possible output values, the range narrows down the actual set of values that the function can attain based on the specific function and its domain.
• Describe how the range of different types of functions, such as linear, quadratic, and absolute value functions, can be characterized.
  • The range of a linear function is an unbounded interval, as the function can produce any real number as an output. The range of a quadratic function is a bounded interval, as the function has a maximum or minimum value. The range of an absolute value function is a non-negative interval, as the absolute value of any number is always greater than or equal to zero. These differences in the range of various function types are due to the inherent characteristics and properties of the functions, which determine the set of output values they can produce.
• Analyze how the transformation of a function, such as a shift, stretch, or reflection, can affect the range of the function.
  • Transformations of functions, such as shifts, stretches, and reflections, can significantly impact the range of the function. A vertical shift of the function can change the minimum or maximum values of the range, while a horizontal shift can affect the input values that produce the minimum and maximum outputs. Stretching or compressing the function vertically can alter the range by changing the distance between the minimum and maximum values. Reflecting the function about the $x$-axis or $y$-axis can also result in a change in the range, as the output values may be transformed to a different interval. Understanding how transformations affect the range is crucial in analyzing the behavior and properties of functions.

Related terms

Domain: The domain of a function is the set of all possible input values for which the function is defined.

Codomain: The codomain of a function is the set of all possible output values that the function can produce.

Absolute Value: The absolute value of a number is the distance of that number from zero on the number line, regardless of its sign.