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1. Relations
A relation is any set of ordered pairs
Definition
DOMAINRANGE
independent variable dependent variable
2. Definition of a 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: set of inputs for a functionRange: set of outputs for a function.
82 2 xxy
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
3. Test if a relation is a function:
Test Algebraically or Test with
Vertical Line Test2xy
x y 2
16)4()2( 22 yx
1)
2)
3)
tells us to apply the rule to a “number” xargument (independent variable)
Function Notation Equation Notation
4. Function Notation
)(xf)(xf
f
)3(f
function name
32)( 2 xxxf 322 xxy
Output is in Range
5. Function as a machine
Example
Inputfrom Domain
)(xfx12)( 2 xxf
)0(f
)1( xf
)( xf
)2( xf
Example: 1. Subtract
2. Add
3. Multiply
4. Divide
6. Constructing Functions
)()())(( xfxgxfg Algebraic combinations of functions to form a new function.
)()())(( xgxfxgf
)()())(( xgxfxfg
)(
)()(
xg
xfx
g
f
Study Tip: Cannot split the argumentComposition Functions are not multiplication ))(( xgf
)()()( hfxfhxf
4)(
13)(2
xxf
xxg
7. Domain
Example: State the domain : 1
1)(
x
xf
Definition
Are there any x-values that would make f(x) not real?
Set-builder notation: Domain: 1| xx
Interval notation Domain: ,11,
In words: Domain: 1x
Domain: The largest set of real numbers for which
f(x) is a real number
7. a) Examples: Finding the Domain
f (x)x 2 3x 1
f (x)2x
3x 5x 2
85)( xxf
Polynomial
Rational
Radical (Square Root)
Function Type DomainExample
9. Difference Quotient
The Difference Quotient:
f (x h) f (x)h
Tells us the rate of change of a function.
1)
2) 132)( 2 xxxf
43)( xxf