Functions and Their Properties

Relations, Functions, Domain, and Range

A relation is any set of ordered pairs. A function is a 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 which the function is defined.

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

• Determining a Function: A relation is a function if no x-value is paired with more than one y-value.

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

Equations Defining Functions

Given an equation in x and y, y is a function of x if for every x in the domain, there is only one corresponding y.

• Test by solving for y in terms of x and checking for uniqueness.

Example: defines y as a function of x. does not, since each x (except 0) has two y-values.

Finding Domains

• From a formula: Exclude x-values that cause division by zero or negative square roots.

• From a graph: The domain is the set of all x-values covered by the graph.

Example: For , the domain is all real numbers except .

Evaluating Functions

• Substitute the given value into the function's formula or read the value from the graph.

Example: If , then .

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 graph of passes the vertical line test; the graph of a circle does not.

Piecewise Functions

• Defined by different expressions for different intervals of the domain.

• Evaluate by determining which interval the input belongs to, then using the corresponding formula.

Example:

Increasing/Decreasing Intervals and Extrema

• A function is increasing on intervals where its graph rises as x increases.

• Decreasing where the graph falls as x increases.

• Relative minimum/maximum: The lowest/highest point in a local region of the graph.

Example: The vertex of is a relative (and absolute) minimum.

Even and Odd Functions

• 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.

Difference Quotient

• The difference quotient is used to measure the average rate of change of a function over an interval.

Example: For , the difference quotient is .

Transformations of Functions

• Shifts, reflections, stretches, and compressions applied to basic functions.

• Graph related functions by applying these transformations to the graph of a standard function.

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

Operations on Functions

• Sum:

• Difference:

• Product:

• Quotient: ,

• Domain: Intersection of the domains of and (for quotient, exclude where )

Composition of Functions

• Domain: All x in the domain of such that is in the domain of

Example: If and , then .

Inverse Functions

• Two functions and are inverses if and for all x in their domains.

• Horizontal line test: A function has an inverse if every horizontal line intersects its graph at most once.

• To find the inverse: Solve for in terms of , then interchange $x$ and $y$.

Example: has inverse .

Complex Numbers and Quadratic Equations

Operations with Complex Numbers

• Complex number: , where

• Sum:

• Difference:

• Product:

• Quotient:

Example:

Square Roots of Negative Numbers

• for

Example:

Solving Quadratic Equations

• By factoring: Set equation to zero, factor, and solve for x.

• By quadratic formula:

Example: Solve by factoring: , so or .

Parabolas and Quadratic Functions

• Standard form:

• Vertex:

• Y-intercept: Set

• X-intercepts: Solve

Example: For , vertex at , ; y-intercept at ; x-intercepts at .

Finding Quadratic Functions from Conditions

• Use given points or vertex/intercept information to set up equations and solve for coefficients.

Example: Find passing through (0,1), (1,2), (2,5).

Optimization Problems

• Model the situation with a quadratic function and find the maximum or minimum value (vertex).

Example: Maximize area given a fixed perimeter.

Polynomial Functions and Their Properties

End Behavior of Polynomials

• As , the leading term determines the end behavior.

• If degree is even and leading coefficient positive, both ends up; if negative, both ends down.

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

Zeros of Polynomials and Graph Behavior

• Find zeros by solving .

• Test points between zeros to determine if the graph is above or below the x-axis.

Example: For , zeros at and .

Linear Factorization Theorem

• Every polynomial of degree can be written as , where are zeros (real or complex).

Example:

Polynomial Division

• Long division: Divide polynomials as with numbers.

• Synthetic division: Shortcut for dividing by .

Example: Divide by using synthetic division.

Remainder and Factor Theorems

• Remainder Theorem: The remainder when is divided by is .

• Factor Theorem: is a factor of if and only if .

Example: If , then is a factor of .

Rational Root Theorem

• Possible rational roots of are , where divides and divides .

Example: For , possible rational roots are .

Solving Polynomial Equations

• Find one zero, factor or divide, then solve the reduced equation for remaining zeros.

Example: If is a zero, divide by and solve the quadratic factor.

Summary Table: Key Concepts and Theorems

Additional info: This guide covers all major topics listed for Exam 1, including function properties, operations, inverses, complex numbers, quadratics, and polynomial theorems. For detailed examples and practice, refer to your textbook and homework assignments.