Functions and Their Properties

Definitions and Key Concepts

Understanding the properties and types of functions is fundamental in precalculus. This section covers the basic definitions, classifications, and behaviors of functions.

• Function: A relation in which each input (from the domain) is assigned exactly one output (in the range).

• Even Function: A function is even if for all in the domain. Its graph is symmetric about the y-axis.

• Odd Function: A function is odd if for all in the domain. Its graph is symmetric about the origin.

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

• Intercepts: Points where the graph crosses the axes. The y-intercept occurs where .

Example: The reciprocal function is neither even nor odd, as and only for .

Polynomial Functions

Structure and Properties

Polynomial functions are algebraic expressions involving sums of powers of with real coefficients. They are classified by degree and can be analyzed for roots, end behavior, and symmetry.

• General Form:

• Degree: The highest power of with a nonzero coefficient.

• Roots/Zeros: Values of for which .

• Multiplicity: The number of times a root is repeated. If is a factor, is a root of multiplicity .

• End Behavior: Determined by the leading term .

Example: For , the vertex is at , and the graph opens upwards.

Factor Theorem

The Factor Theorem states that is a factor of if and only if .

• Application: To solve , factor the polynomial and set each factor equal to zero.

Example: If and is a factor, then .

Symmetry of Functions

Even, Odd, and Neither

Determining the symmetry of a function helps in graphing and understanding its properties.

• Even Function:

• Odd Function:

• Neither: If neither condition is satisfied.

Example: is even; is odd; is neither.

Rational Functions

Definition and Properties

A rational function is a function of the form , where and are polynomials and .

• Domain: All real numbers except where .

• Vertical Asymptotes: Occur at values of where and .

• Horizontal Asymptotes: Determined by the degrees of and .

• Holes: Occur where and have a common factor.

• Slant (Oblique) Asymptotes: Occur if the degree of is exactly one more than the degree of .

Example: has a vertical asymptote at and a horizontal asymptote at as .

Table: Asymptotes and Holes in Rational Functions

Additional info: Table entries inferred for general rational function analysis.

Polynomial Division

Long Division and Synthetic Division

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

• Long Division: Similar to numerical long division, used for dividing polynomials of any degree.

• Synthetic Division: A shortcut for dividing by linear factors of the form .

Example: To solve given that $2-2$ are roots, factor and use division to find remaining roots.

Graphing Functions

Vertex, Range, and Solution Set

Graphing quadratic and rational functions involves identifying key features such as the vertex, range, and zeros.

• Vertex (for quadratics): The point in .

• Range: The set of possible output values, often determined by the vertex and direction of opening.

• Solution Set: The set of -values where .

Example: For , the vertex is at , so the vertex is .

Applications of Polynomial Models

Modeling with Polynomials

Polynomials can be used to model real-world data, such as population growth or economic trends. However, the usefulness of a polynomial model depends on the context and the time period considered.

• Extrapolation: Using a model to predict values outside the range of observed data can lead to inaccuracies, especially with high-degree polynomials.

• Interpretation: Always consider the domain and range relevant to the real-world situation.

Example: If models thefts, it may not be reliable for years far from the data used to create the model.