Functions and Their Properties

Piecewise Functions

Piecewise functions are defined by different expressions depending on the input value. They are useful for modeling situations where a rule changes based on the domain.

• Definition: A function defined by multiple sub-functions, each applying to a certain interval of the domain.

• Example: If for and for , then:

• Domain: The set of all values for which the function is defined.

Function Operations and Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and is a precursor to calculus.

• Difference Quotient:

• Example: For :

• Application: Used to find the slope of a function at a point.

Polynomial and Rational Functions

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. This is essential for solving equations and simplifying expressions.

• Common Methods: Factoring out the greatest common factor, grouping, and using formulas like difference of squares.

• Example:

• Difference of Squares:

• Example:

Factoring by Substitution

Sometimes, substitution is used to simplify factoring, especially for higher-degree polynomials.

• Example: Let , then factor as .

Domain of Rational Functions

The domain of a rational function excludes values that make the denominator zero.

• Example: For , denominator is zero when .

• Domain: All real numbers except and .

Trigonometric Functions and Identities

Trigonometric Values and Quadrants

Trigonometric functions have different signs depending on the quadrant.

• Cosine: Positive in Quadrants I and IV, negative in II and III.

• Sine: Positive in Quadrants I and II, negative in III and IV.

• Tangent: Positive in Quadrants I and III, negative in II and IV.

Trigonometric Identities

Identities are equations involving trigonometric functions that are true for all values in their domains.

• Sum and Difference Formulas:

• Double Angle Formula:

• Pythagorean Identity:

Solving Trigonometric Equations

Solving trigonometric equations often involves factoring, using identities, and considering the domain and range of the functions.

• Example: Solve

• Example: Solve

• Set each factor to zero: or

• Solutions: or

Application: Finding Exact Values

Using sum and difference formulas to find exact values for trigonometric functions.

• Example:

Summary Table: Trigonometric Signs by Quadrant

Additional info:

• Some steps and context were inferred from fragmented notes, such as domain restrictions and factoring methods.

• Trigonometric equation solutions were expanded for clarity.

• Examples were reconstructed for completeness.