Functions and Their Properties

Introduction to Functions

Functions are foundational concepts in calculus and are used to model relationships between variables in business, economics, and the sciences. Understanding the properties and types of functions is essential for further study in calculus.

• Function: A rule that assigns to each value of a real variable x exactly one value of another real variable y.

• Independent variable: The variable x in the function y = f(x).

• Dependent variable: The variable y in the function y = f(x).

• Function notation: y = f(x) indicates that y is a function of x.

Example: If f(x) = x^2 + 4x + 3, then f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1.

Types of Numbers and the Number Line

Rational and Irrational Numbers

• Rational Number: A number that can be written as a fraction m/n where m and n are integers and n ≠ 0. Its decimal representation is either terminating or repeating. Example: (repeats),

• Irrational Number: A number with a non-repeating, non-terminating decimal expansion. Example:

The Number Line: A geometric representation of real numbers, where each point corresponds to a real number.

Intervals on the Real Line

Finite, Open, and Closed Intervals

Domain of a Function

Definition and Examples

• Domain: The set of all acceptable values for the variable x (all possible inputs for the function).

Graphs of Functions

Graph and the Vertical Line Test

• Graph of a Function: The set of all points where is in the domain of ; typically forms a curve in the -plane.

• Vertical Line Test: A curve is the graph of a function if and only if no vertical line intersects the curve more than once.

Important Types of Functions

Linear, Quadratic, Polynomial, Rational, Power, and Absolute Value Functions

• Linear Function: Graph is a straight line.

• Quadratic Function: Graph is a parabola.

• Polynomial Function: Where is a nonnegative integer.

• Rational Function: Where and are polynomials and .

• Power Function: Where is a real number.

• Absolute Value Function: Defined as if , if .

The Algebra of Functions

Operations on Functions

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

Example: If and , then can be combined into a single rational function.

Composition of Functions

Definition and Application

• Composition:

Example: If (British to French hat size) and (French to US size), then converts British to US hat size.

Zeros of Functions

Definition and Finding Zeros

• Zero of a Function: A value such that .

• For quadratic functions , the quadratic formula gives the zeros:

Example: For , , , :

So or .

Intersection of Functions

Finding Points of Intersection

• Set the two functions equal: .

• Solve for (may require the quadratic formula).

• Substitute back into either function to find .

Example: If and , set and solve for .

Factoring and Solving Equations

Factoring Techniques

• Factor out common terms.

• Use patterns such as .

• Set each factor equal to zero to solve for .

Exponents and Their Properties

Definition and Laws of Exponents

• Exponent: Indicates how many times a number (the base) is multiplied by itself.

• Laws of Exponents:

  • (if )

Example:

Geometric Applications

Rectangular Corral and Box Problems

• Assign variables to dimensions (e.g., and for length and width).

• Area of a rectangle:

• Volume of a box:

• Surface area of an open box:

Example: For a rectangular corral with area 2500 sq ft, .

Example: For a box with dimensions , , , the volume is and the surface area (open top) is .