Section 2.1 Functions

1. Relations A relation is any set of ordered pairs Definition DOMAINRANGE independent variable dependent variable

2. Definition of a function Function: Function: a relation where: each element in domain corresponds to EXACTLY one element in the range. Function: Function: a relation where: each element in domain corresponds to EXACTLY one element in the range. {(2,6),(-3,6),(4,9),(2,10)} Definitions Examples of functions. Domain Domain: set of inputs for a function Range Range: set of outputs for a function.

3. Functions as Equations Determine if an equation is a function. 1.Solve for y. 2.If each x is associated with a unique y, then function Goal: Method 1: Algebraically Method 2: Graphically – Vertical Line Test 1.Graph the equation. 2.If no vertical line intersects the graph more than once, then function

Test 3. Test if a relation is a function: or Test Algebraically or Test with Vertical Line Test or Test Algebraically or Test with Vertical Line Test 1) 2) 3)

tells us to apply the rule to a “number” x argument (independent variable) Function Notation Equation Notation 4. Function Notation function name

Range Output is in Range 5. Function as a machine Example Input fromDomain

Example: 1.Subtract 2.Add 3.Multiply 4.Divide 6. Constructing Functions Algebraic combinations of functions to form a new function. Study Tip: Cannot split the argument Composition Functions are not multiplication Study Tip: Cannot split the argument Composition Functions are not multiplication

7. Domain Example: State the domain : Definition Are there any x-values that would make f(x) not real? Set-builder notation: Domain: Interval notation Domain: In words: Domain: Domain : The largest set of real numbers for which f(x) is a real number

7. a) Examples: Finding the Domain Polynomial Rational Radical (Square Root) Polynomial Rational Radical (Square Root) Function Type Domain Example

9. Difference Quotient The Difference Quotient: Tells us the rate of change of a function. 1) 2)

tells us to apply the rule for to a “number” x is a symbol for the function (could be any letter) is the argument (independent variable) The value of is the y coordinate on the graph. 2.2 The Graph of