Determining the Output Values of a Function

Identifying the Codomain

The codomain of a function represents the set of all possible output values. Understanding the codomain is a crucial first step in determining the function's range.

Defining the Range

The range, a subset of the codomain, comprises the actual output values generated by the function when it acts upon every element within its domain. It's the set of all values the function can attain.

Methods for Determining the Range

Graphical Analysis

For functions represented graphically, the range can be visually determined by observing the y-values (vertical axis) spanned by the graph. The lowest and highest y-values, inclusive of any values in between, represent the range's extent.

Algebraic Techniques

Analytical methods involve manipulating the function's equation to solve for the dependent variable in terms of the independent variable. This allows determining the boundaries of the output values. Techniques employed often include solving inequalities, considering the domain restrictions, and analyzing the behavior of the function (e.g., asymptotes, limits).

Set Notation and Interval Notation

The range is formally expressed using set notation or interval notation, concisely representing the boundaries of the output values. Set notation lists the elements, while interval notation utilizes parentheses or brackets to indicate whether the endpoints are included.

Examples

Linear Functions

The range of a linear function, unless otherwise restricted by the domain, is typically the set of all real numbers (ℝ).

Quadratic Functions

The range of a quadratic function depends on its vertex's y-coordinate and whether it opens upwards or downwards. For upward-opening parabolas, the range is defined by y-values greater than or equal to the vertex's y-coordinate. The reverse is true for downward-opening parabolas.

Trigonometric Functions

The ranges of trigonometric functions are periodic and bounded, typically within specific intervals. For example, the sine and cosine functions have a range of [-1, 1].

Piecewise Functions

The range of a piecewise function is determined by analyzing the range of each piece and combining them into a single set that accounts for all possible output values.