Section 1.1: Functions

Objective 1: Describe a Relation

A relation is a correspondence between two sets, typically called the domain (input values) and the range (output values). In a relation, each element from the domain is paired with one or more elements from the range.

• Domain: The set of all possible input values (usually x-values).

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

There are several ways to express a relation:

1. Using words

2. Using a table of numbers or a set of ordered pairs

3. Plotting points or mapping

4. Using an equation

Example: A scientist records the average high temperature in Los Angeles for the first five months of the year. The relation can be expressed as:

• Domain: {January, February, March, April, May}

• Range: {68°F, 69°F, 70°F, 73°F, 74°F}

• Ordered pairs: (January, 68°F), (February, 69°F), (March, 70°F), (April, 73°F), (May, 74°F)

• Mapping: Each month maps to its corresponding temperature.

Objective 2: Determine Whether a Relation Represents a Function

A function is a special type of relation in which each element of the domain is paired with exactly one element of the range. In other words, for every input, there is only one output.

• The set X is called the domain of the function.

• The set Y is called the range of the function.

• The set of all images of the elements in the domain is called the range of the function.

Key Point: It is not allowed for one input to correspond to more than one output in a function.

Example: Determining if a relation is a function:

• Person to Birthday: Each person has only one birthday, so this is a function.

• Father to Daughter: A father can have more than one daughter, so this is not a function.

Vertical-Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Objective 3: Use Function Notation; Find the Value of a Function

Function notation is a way to name functions and indicate the input and output. If y is a function of x, we write , where f is the name of the function, x is the input, and f(x) is the output.

• f(x): Read as "f of x," represents the value of the function at x.

• The variable y is called the dependent variable because its value depends on x.

• The variable x is called the independent variable.

Example: For , evaluate:

Objective 4: Find the Difference Quotient of a Function

The difference quotient of a function f at x is given by:

• The difference quotient is used in calculus to define the derivative.

Example: For ,

• Difference quotient:

Objective 5: Find the Domain of a Function Defined by an Equation

The domain of a function defined by an equation is the set of all real numbers for which the equation is defined (i.e., produces real outputs).

• If the equation has a denominator, exclude values that make the denominator zero.

• If the equation has a square root, exclude values that make the radicand negative.

Example: For , the domain is all real numbers except and .

General Steps to Find the Domain:

• Start with all real numbers.

• Exclude values that make any denominator zero.

• Exclude values that make any even root negative.

Objective 6: Find the Sum, Difference, Product, and Quotient of Two Functions

Given functions f and g, we can define new functions as follows:

• Sum:

• Difference:

• Product:

• Quotient:

The domain of each new function is the intersection of the domains of f and g, with additional restrictions for the quotient (where ).

Example: If and , then:

• Domain:

Section 1.2: Graphs of Functions

Objective 1: Identify the Graph of a Function

The graph of a function is the set of all points in the plane such that for each in the domain.

• Not every collection of points in the plane is the graph of a function.

• The vertical-line test states that a graph is a function if and only if every vertical line intersects the graph at most once.

Example: The graph of passes the vertical-line test and is a function.

Objective 2: Obtain Information from or about the Graph of a Function

Given the graph of a function, you can determine:

• The value of the function at a given x (i.e., is the y-value at ).

• The domain and range of the function.

• The intercepts (where the graph crosses the axes).

• Where the function is positive or negative.

• Where the function attains certain values.

Example: If the graph passes through , then .

Table: Summary of Function Operations

Additional info: These notes cover foundational concepts in College Algebra, including relations, functions, function notation, domains, operations on functions, and interpreting graphs. The difference quotient is introduced as a precursor to calculus.