Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions depending on the input value (domain). Each 'piece' applies to a specific interval of the independent variable.

• Definition: A function composed of two or more sub-functions, each with its own domain.

• Example:

• Graphing: Plot each piece on its respective interval, using open or closed circles to indicate whether endpoints are included.

Evaluating Piecewise Functions

To find , determine which interval falls into and use the corresponding expression.

• Example: For above, (since ), (since ).

Intervals of Increase, Decrease, and Constancy

Analyze each piece to determine where the function is increasing, decreasing, or constant.

• Increasing: Function values rise as increases.

• Decreasing: Function values fall as increases.

• Constant: Function values remain unchanged as increases.

• Example: For above, both pieces are constant on their intervals.

Function Operations and Composition

Basic Operations

Functions can be added, subtracted, multiplied, or divided to create new functions.

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

Function Composition

The composition of functions, denoted or , means applying first, then to the result.

• Example: If and , then

Domain of Combined Functions

The domain of a combined function is the set of all values for which the operation is defined.

• For addition, subtraction, multiplication: Intersection of the domains of and .

• For division: Intersection of domains, excluding values where .

• For composition: must be in the domain of , and must be in the domain of .

Transformations of Functions

Types of Transformations

Transformations change the position or shape of a function's graph.

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

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

• Reflections: reflects across the -axis; reflects across the -axis.

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

Examples of Transformations

• Quadratic: , (vertical shift up by 4)

• Square Root: , (reflection over -axis, shift down by 2)

• Reciprocal: , (vertical shift down by 3)

• Absolute Value: , (horizontal shift left by 2, vertical stretch by 3, shift down by 1)

Variation Problems

Direct and Inverse Variation

Variation describes how one quantity changes in relation to another.

• Direct Variation: (as increases, increases proportionally)

• Inverse Variation: (as increases, decreases proportionally)

• Variation with Powers: varies inversely as the square of :

• Variation with Roots: varies directly as the square root of :

Solving Variation Problems

• Find the constant : Substitute given values into the variation equation and solve for .

• Use to solve for unknowns: Substitute and the new value of to find .

• Example: If varies inversely as , and when , then , so .

Summary Table: Types of Function Transformations

Additional info: The study notes above expand on the original questions by providing definitions, formulas, and examples for each concept, ensuring a self-contained guide for College Algebra students.