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 where 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.

Behavior and Symmetry of Functions

• Increasing Function: f(x) increases as x increases on an interval.

• Decreasing Function: f(x) decreases as x increases on an interval.

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

• Symmetry:

  • x-axis: Graph is symmetric if replacing y with -y yields the same equation.

  • y-axis: Graph is symmetric if replacing x with -x yields the same equation.

  • Origin: Graph is symmetric if replacing (x, y) with (-x, -y) yields the same equation.

• Even Function: for all x in the domain (symmetric about the y-axis).

• Odd Function: for all x in the domain (symmetric about the origin).

Example: is even; is odd.

Piecewise Functions

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

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

Example:

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 Shift: shifts the graph up (if ) or down (if ).

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

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

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

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

Example: is shifted right 2 units and up 3 units.

Combinations and Compositions of Functions

Function Operations

• Sum:

• Difference:

• Product:

• Quotient: ,

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

Composite Functions

• Definition:

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

• Decomposition: Express a function as a composition of two or more 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 a function satisfies and .

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

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

Example: has inverse .

Complex Numbers

Definition and Operations

• Imaginary Unit:

• Standard Form: , where and are real numbers.

• Addition/Subtraction:

• Multiplication:

• Complex Conjugate: The conjugate of is .

• Division:

Roots and Quadratic Equations

• To find , write as .

• Quadratic equations with negative discriminant have complex conjugate 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 at vertex); if , opens downward (maximum at vertex).

Example: has vertex at and opens upward.

Polynomial Functions and Their Graphs

Definition and Properties

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

• Degree: The highest power of .

• Leading Coefficient: The coefficient of the highest degree term.

End Behavior (Leading Coefficient Test)

• If degree is even and leading coefficient is positive, both ends up.

• If degree is even and leading coefficient is negative, both ends down.

• If degree is odd and leading coefficient is positive, left end down, right end up.

• If degree is odd and leading coefficient is negative, left end up, right end down.

Zeros of Polynomial Functions

• Zeros: Values of where (also called roots or x-intercepts).

• Find zeros by factoring the polynomial and setting each factor equal to zero.

Example: has zeros at and .