Precalculus Study Guide: Functions, Polynomials, Rational Functions, and Symmetry

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Functions and Their Properties

Definitions and Types of Functions

Functions are mathematical relationships that assign each input exactly one output. Understanding the properties and classifications of functions is fundamental in precalculus.

Even Function: A function f(x) is even if f(-x) = f(x) for all x in its domain.
Odd Function: A function f(x) is odd if f(-x) = -f(x) for all x in its domain.
Neither: If a function does not satisfy either condition, it is classified as neither even nor odd.

Example: The function f(x) = x^2 is even because f(-x) = (-x)^2 = x^2 = f(x).

Function Operations

Functions can be combined using addition, subtraction, multiplication, and division. The composition of functions involves applying one function to the result of another.

Sum:
Difference:
Product:
Quotient: , where
Composition:

Example: If and , then .

Polynomials

Polynomial Functions

Polynomial functions are expressions involving sums of powers of x with real coefficients. They are classified by degree and can be factored and analyzed for roots.

General Form:
Degree: The highest power of x with a nonzero coefficient.
Roots/Zeros: Values of x for which f(x) = 0.
Multiplicity: The number of times a particular root occurs.

Example: has roots (multiplicity 2) and (multiplicity 1).

Factor Theorem

The Factor Theorem states that if , then is a factor of .

Application: Used to find all roots of a polynomial when one root is known.

Example: If and is a root, then is a factor.

Polynomial Division

Polynomial division is used to divide one polynomial by another, often to simplify rational expressions or find roots.

Long Division: Similar to numerical long division, used for dividing polynomials.
Synthetic Division: A shortcut method for division when the divisor is linear.

Quadratic and Rational Functions

Quadratic Functions

Quadratic functions have the form . Their graphs are parabolas.

Vertex: The point where the parabola reaches its maximum or minimum.
Range: The set of possible output values.
Solution Set: Values of x where .

Vertex Formula:

Example: For , the vertex is at .

Rational Functions

Rational functions are quotients of polynomials. Their graphs may have asymptotes and holes.

Vertical Asymptote: Occurs where the denominator is zero and the numerator is nonzero.
Horizontal Asymptote: Determined by the degrees of numerator and denominator.
Slant Asymptote: Occurs when the degree of the numerator is one more than the denominator.
Hole: Occurs where both numerator and denominator are zero for the same value of x.

Example: has a hole at .

Symmetry of Functions

Even, Odd, and Neither Functions

Determining symmetry helps in graphing and understanding function behavior.

Even: Symmetric about the y-axis.
Odd: Symmetric about the origin.
Neither: No symmetry.

Example: is odd; is even.

Graphing Functions

Graphing Quadratic and Rational Functions

Graphing involves plotting key points, identifying intercepts, and asymptotes.

Quadratic: Identify vertex, axis of symmetry, and intercepts.
Rational: Identify holes, vertical/horizontal/slant asymptotes, and intercepts.

Example: For , there is a vertical asymptote at .

Application of Polynomials

Modeling with Polynomials

Polynomials can be used to model real-world phenomena, such as population growth or economic trends. The validity of a model depends on the context and time period.

Example: If models thefts over time, consider whether the model remains accurate for extended periods.

Summary Table: Asymptotes and Holes in Rational Functions

Hole	Vertical Asymptote	Horizontal Asymptote	Slant Asymptote
Find where numerator and denominator both zero (Additional info: Set and )	Set denominator zero, solve	Degree numerator = degree denominator, so horizontal asymptote at	None
None (numerator never zero when denominator zero)	Set denominator zero, gives and	Degree numerator > denominator, so slant asymptote exists	Find by polynomial division
None (numerator never zero when denominator zero)	Set denominator zero, gives and	Degree numerator < denominator, so horizontal asymptote at	None

Key Formulas

Vertex of Quadratic:
Range of Quadratic: For , ; for , where is the vertex y-value.
Rational Function Asymptotes: Vertical: Set denominator zero; Horizontal: Compare degrees; Slant: If numerator degree is one more than denominator.

Additional info:

Some questions require graphing and analysis of specific functions, which is a core skill in precalculus.
Real-world modeling with polynomials is an application topic, emphasizing the importance of context in mathematical modeling.