2.3: Functions Objectives: To determine if a relation is a function To find the domain and range of a function To evaluate functions

Vocabulary As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook. Relation Function Input Output Domain Range Set-Builder Notation Interval Notation Function Notation

Relations A mathematical relation is the pairing up (mapping) of inputs and outputs.

Relations A mathematical relation is the pairing up (mapping) of inputs and outputs. Domain: the set of all input values Range: the set of all output values

Calvin and Hobbes! A toaster is an example of a function. You put in bread, the toaster performs a toasting function, and out pops toasted bread.

Calvin and Hobbes! “What comes out of a toaster?” “It depends on what you put in.” You can’t input bread and expect a waffle!

What’s Your Function? A function is a relation in which each input has exactly one output. A function is a dependent relation Output depends on the input Relations Functions

What’s Your Function? A function is a relation in which each input has exactly one output. Each output does not necessarily have only one input Relations Functions

How Many Girlfriends? If you think of the inputs as boys and the output as girls, then a function occurs when each boy has only one girlfriend. Otherwise the boy gets in BIG trouble. Darth Vadar as a “Procurer.”

What’s a Function Look Like?

What’s a Function Look Like?

What’s a Function Look Like?

What’s a Function Look Like?

Exercise 1a Tell whether or not each table represents a function. Give the domain and range of each relationship.

Exercise 1b The size of a set is called its cardinality. What must be true about the cardinalities of the domain and range of any function?

Exercise 2 Which sets of ordered pairs represent functions? {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)} {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)} {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

Break this Function!!!  

Exercise 3 Which of the following graphs represent functions? What is an easy way to tell that each input has only one output?

Vertical Line Test A relation is a function iff no vertical line intersects the graph of the relation at more than one point If it does, then an input has more than one output. Function Not a Function

Functions as Equations To determine if an equation represents a function, try solving the thing for y. Make sure that there is only one value of y for every value of x.

Exercise 4  

Set-Builder Notation Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation. Examples: {x | x < -2} reads “the set of all x such that x is less than negative 2”.

Set-Builder Notation Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation. Examples: {x : x < -2} reads “the set of all x such that x is less than negative 2”.

Interval Notation Another way to describe an infinite set of numbers is with interval notation. Parenthesis indicate that first or last number is not in the set: Example: (-, -2) means the same thing as x < -2 Neither the negative infinity or the negative 2 are included in the interval Always write the smaller number, bigger number

Interval Notation Another way to describe an infinite set of numbers is with interval notation. Brackets indicate that first or last number is in the set: Example: (-, -2] means the same thing as x  -2 Infinity (positive or negative) never gets a bracket Always write the smaller number, bigger number

Domain and Range: Graphs Domain: All x-values (L → R) {x: -∞ < x < ∞} Range: All y-values (D ↑ U) {y: y ≥ -4} Range: Greater than or equal to -4 Domain: All real numbers

Exercise 5 Determine the domain and range of each function.

Domain and Range: Equations Domain: What you are allowed to plug in for x. Easier to ask what you can’t plug in for x. Limited by division by zero or negative even roots Can be explicit or implied Range: What you can get out for y using the domain. Easier to ask what you can’t get for y.

Exercise 6a    

Exercise 6b      

Dependent Quantities Functions can also be thought of as dependent relationships. In a function, the value of the output depends on the value of the input. Independent quantity: Input values, x-values, domain Dependent quantity: Output value, which depends on the input value, y-values, range