Functions and Their Properties

Definition of a Function

A function is a relation that assigns each element in the domain to exactly one element in the range. Functions are often represented as , where is the input variable.

• Domain: The set of all possible input values () for which the function is defined.

• Range: The set of all possible output values () produced by the function.

• Function Notation: denotes the value of the function at .

Example: For , the domain is all real numbers, and the range is all non-negative real numbers.

Evaluating Functions

To evaluate a function, substitute the given value into the function's formula.

• Example: If , then .

Graphing Functions and Transformations

Basic Graphs

Common functions include linear, quadratic, cubic, and absolute value functions. Each has a characteristic graph.

• Linear Function: (straight line)

• Quadratic Function: (parabola)

• Absolute Value Function: (V-shaped graph)

Transformations of Functions

Transformations change the position or shape of a graph. The main types are:

• Vertical Shifts: shifts the graph up () or down ().

• Horizontal Shifts: shifts the graph right () or left ().

• Reflections: reflects the graph over the x-axis; reflects over the y-axis.

• Vertical Stretch/Compression: stretches () or compresses () vertically.

Example: The graph of is a parabola shifted right by 2 units and up by 3 units.

Piecewise Functions

Definition and Evaluation

A piecewise function is defined by different expressions for different intervals of the domain.

• Example:

• To evaluate, determine which interval the input belongs to and use the corresponding formula.

Function Composition

Definition

Composition of functions involves applying one function to the result of another. The notation is .

• Example: If and , then .

Inverse Functions

Definition and Properties

An inverse function reverses the effect of . If , then .

• Finding the Inverse: Solve for in terms of , then interchange and .

• Example: For , set , solve for : , so .

HTML Table: Types of Function Transformations

Additional info:

• Some questions on the file involve identifying domains, evaluating piecewise functions, and describing transformations, which are core Precalculus topics.

• Where the image was unclear, standard Precalculus definitions and examples were provided to ensure completeness.