Functions

Relations and Functions

A relation is a connection between input (x) and output (y) values, often represented as ordered pairs (x, y). A function is a special type of relation where each input has at most one output. To determine if a graph represents a function, use the Vertical Line Test: if any vertical line crosses the graph more than once, it is not a function.

• Key Point: Each x-value must correspond to only one y-value in a function.

• Example: The relation {(−3, 5), (0, 2), (3, 5)} is a function because each x-value is unique.

• Example: The relation {(2, 5), (0, 2), (2, 9)} is not a function because x = 2 corresponds to two different y-values.

Verifying Functions from Equations

To verify if an equation is a function, solve for y in terms of x. If y has an even power or the equation is a circle, it is not a function. Functions can be written in function notation: replace y with f(x).

• Key Point: If an equation has an even power of y, it is not a function.

• Example: y = 3x − 4 can be written as f(x) = 3x − 4.

Domain and Range

Finding the Domain and Range of a Graph

The domain of a function is the set of allowed x-values, and the range is the set of allowed y-values. To find the domain, project the graph onto the x-axis; to find the range, project onto the y-axis. Use interval notation or set builder notation to express answers.

• Key Point: Use brackets [ ] for values included, parentheses ( ) for values not included.

• Union Symbol: Use ∪ to combine multiple intervals.

Finding the Domain of an Equation

When given an equation, determine domain restrictions by identifying values that make the function undefined. Common restrictions include:

• Square Roots: The inside of the square root must be non-negative.

• Fractions: The denominator must not be zero.

Example: For , the domain is because x must be non-negative.

Example: For , the domain is because x cannot be 5.

Transformations of Functions

Types of Transformations

Transformations change the appearance or position of a function's graph. The main types are:

• Reflection: Flips the graph over the x-axis or y-axis.

• Shift: Moves the graph horizontally or vertically.

• Stretch/Shrink: Changes the graph's shape by multiplying by a constant.

• Reflection: reflects over the x-axis.

• Shift: shifts the graph h units horizontally and k units vertically.

• Stretch: stretches or shrinks the graph vertically.

Effects of Transformations

• Vertical Shift: moves the graph up if k > 0, down if k < 0.

• Horizontal Shift: moves the graph right if h > 0, left if h < 0.

• Vertical Stretch/Shrink: stretches if , shrinks if .

• Horizontal Stretch/Shrink: shrinks if , stretches if .

Function Operations

Adding, Subtracting, Multiplying, and Dividing Functions

Functions can be combined using addition, subtraction, multiplication, and division. The domain of the resulting function is the set of values common to the domains of the original functions, with additional restrictions for division (denominator ≠ 0).

• Example:

• Example:

• Example: , domain excludes values where

Function Composition

Composing Functions

Function composition involves substituting one function into another. The notation means . The domain of the composite function is restricted by the domains of both functions.

• Example: If and , then

Decomposing Functions

Decomposition is the reverse of composition: expressing a function as a combination of two or more simpler functions.

• Example: can be written as where and