Functions and Their Graphs

Relations and Functions

A relation is any set of ordered pairs (x, y). A function is a special type of relation in which each input (x-value) corresponds to exactly one output (y-value).

• Domain: The set of all possible input values (x-values) for the relation or function.

• Range: The set of all possible output values (y-values).

• Interval Notation: Used to describe domains and ranges, e.g., .

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: The set { (1,2), (2,3), (3,4) } is a function, but { (1,2), (1,3) } is not.

Increasing, Decreasing, and Constant Functions

A function is:

• Increasing on an interval if, as x increases, f(x) increases.

• Decreasing on an interval if, as x increases, f(x) decreases.

• Constant on an interval if f(x) remains the same as x increases.

Example: is decreasing on and increasing on .

Symmetry of Graphs

• Symmetry with respect to the x-axis: Replacing y with -y yields the same equation.

• Symmetry with respect to the y-axis: Replacing x with -x yields the same equation.

• Symmetry with respect to the origin: Replacing x with -x and y with -y yields the same equation.

Even Functions: Satisfy (symmetric about the y-axis). Odd Functions: Satisfy (symmetric about the origin).

Example: is even; is odd.

Piecewise Functions

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

• Definition:

• To graph, plot each piece on its specified interval.

Example: See above definition.

Transformations of Graphs

Basic Functions and Their Graphs

Know the domain, range, and graph for these seven basic functions:

• (Identity function)

• (Constant function)

• (Absolute value function)

• (Quadratic function)

• (Square root function)

• (Cubic function)

• (Cube root function)

Types of Transformations

• Vertical Shifts: shifts the graph up by k units; shifts down.

• Horizontal Shifts: shifts left by h units; shifts right.

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

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

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

Example: To graph , shift right 2 units and up 3 units.

Combinations and Compositions of Functions

Function Operations

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

• Domain: The domain of the combination is the intersection of the domains of and (for division, exclude where ).

Composite Functions

• Definition:

• Domain: x must be in the domain of , and must be in the domain of .

• Decomposition: Express a function as a composition of two functions.

Example: If and , then .

Inverse Functions

One-to-One Functions and the Horizontal Line Test

• A function is one-to-one if each output is produced by exactly one input.

• Horizontal Line Test: A function is one-to-one if no horizontal line intersects its graph more than once.

Inverse Functions

• Definition: The inverse of , denoted , satisfies and .

• Finding the Inverse: Swap x and y in , then solve for y.

• Graphing: The graph of is the reflection of the graph of over the line .

Example: If , then .

Complex Numbers

Definition and Operations

• Imaginary Unit:

• Standard Form: , where and are real numbers.

• Addition/Subtraction: Combine like terms:

• Multiplication: Use distributive property and .

• Complex Conjugate: For , the conjugate is .

• Division: Multiply numerator and denominator by the conjugate of the denominator.

Example:

Roots of Negative Numbers and Quadratic Equations

• To find , write as .

• Quadratic equations with negative discriminant have imaginary solutions.

• Quadratic Formula:

Quadratic Functions

Graphing Quadratic Functions

• Standard Form:

• General Form:

• Vertex: For , the vertex is .

• Axis of Symmetry:

• Minimum/Maximum: If , the parabola opens upward (minimum); if , it opens downward (maximum).

Example: has vertex at (1, 3) and opens upward.

Polynomial Functions and Their Graphs

Definition and End Behavior

• Polynomial Function: , where is a non-negative integer and coefficients .

• End Behavior (Leading Coefficient Test): The degree and leading coefficient determine the behavior as .

Zeros of Polynomial Functions

• Zeros: Values of x where ; these are the x-intercepts of the graph.

• Factoring: To find zeros, factor the polynomial and set each factor equal to zero.

Example: , so zeros are and .