Chapter 1: Functions

1.1 Review of Functions

Understanding functions is essential for calculus. This section introduces the concept of functions, their representations, and foundational properties.

• Relation: Any connection between pairs of numbers. An element from the input can have more than one element in the output.

• Function: A rule that assigns to each value x in a set D a unique value denoted f(x). The set D is the domain of the function. The range is the set of all values of f(x) produced as x varies over the domain.

• Independent variable: Variable associated with the domain (input).

• Dependent variable: Variable associated with the range (output).

• Graph of a function: The set of all points (x, f(x)).

Representations of Functions

• Numerical: Table of values

• Graphical: Graph

• Symbolic: Formula

Vertical Line Test

A graph represents a function if and only if every vertical line intersects the graph at most once. If a vertical line crosses more than once, the graph does not represent a function.

Examples

• Graph the following functions and identify their domains and ranges:

• For y = x^2 + 1: Domain: Range:

• For z = \sqrt{4 - t^2}: Domain: Range:

• For w = \frac{1}{u-1}: Domain: Range:

Function Operations

• Sum/Difference:

• Product:

• Quotient: ,

Composite Functions

Given two functions and , the composite function is defined by:

• Evaluated in two steps: , then .

• The domain of consists of all in the domain of such that is in the domain of .

• Commutative property does not apply: in general.

Secant Line and Difference Quotient

The secant line is a line through two points on a curve. Its slope is called the difference quotient:

• First form (points and ):

• Second form (points and ):

Application: Velocity of a Moving Object

• The position (in meters) of an object as a function of time (in seconds) can be used to calculate average velocity between two points:

• Average velocity between and :

Symmetry in Graphs

• Symmetry about the y-axis: (Even function)

• Symmetry about the x-axis: Not a function (fails vertical line test)

• Symmetry about the origin: (Odd function)

Even and Odd Functions

• Even function: for all in the domain. Graph is symmetric about the y-axis. Example: .

• Odd function: for all in the domain. Graph is symmetric about the origin. Example: .

• Polynomials with only even powers are even; with only odd powers are odd; with both, neither.

1.2 Representing Functions

Functions can be classified into several types, each with unique properties and applications.

Types of Functions

• Polynomial functions: - Degree: highest power of - Leading coefficient: - is a nonnegative integer

• Rational functions: , where and are polynomials and

• Algebraic functions: Functions involving algebraic operations, not expressible as polynomials or rational functions (e.g., )

• Exponential and logarithmic functions: ,

• Trigonometric functions: , , etc.

Roots of Functions

• Roots (zeros): Values of for which .

• A polynomial of degree can have up to roots.

• Quadratic example: has one root at ; has no real roots (imaginary roots).

Linear Functions

• Form: , where is the slope and is the y-intercept.

• Parallel lines have the same slope but different y-intercepts.

• Slope between points and :

• Equation of a line with slope through point :

Piecewise Functions

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

• Example:

Rational Functions

• Form: ,

• Domain: All such that

A Library of Functions

Additional info: This summary covers the foundational concepts of functions, their types, properties, and representations, which are essential for further study in calculus.