Name: _________________________________________
Date: _____________________
4.1 Notes
Properties of Functions
PreCalc
VOCAB:
 Relation: a set of _____________________________

Function: a relation that _______________________________________________________
TYPES OF FUNCTIONS:
Function Type
LINEAR
QUADRATIC
CUBIC
SQUARE ROOT
CUBE ROOT
ABSOLUTE VALUE
RATIONAL
General Equation
Sketch
Examples
VOCAB:
 Domain: the set of all possible _____________________________________

Range: the set of all possible _____________________________________

Zeros: the ________________ of the ________________ of any function.
(Zeros are sometimes referred to as the ________________)
Examples:
Use the graph to determine the domain, range and zeros of each function.
1.
2.
Domain:
Domain:
Range:
Range:
Zero(s):
Zero(s):
3.
4.
Domain:
Domain:
Range:
Range:
Zero(s):
Zero(s):
*It is easy to determine the domain of a function when given an equation.
 Many domains will be listed as “All Real Numbers” – What does this mean??
 Restrictions in domain may occur depending on the type of function you are dealing with!
THINK: When will my function be undefined?
Domain Restriction Considerations…
1.
Square Root Functions: _______________________________________________________________
2. Rational Functions: ___________________________________________________________________
Examples: Give the domain of each function.
1
3
1. 𝑓(𝑥) =
2. 𝑓(𝑥) = 4 𝑥 − 6
𝑥−7
5
4. 𝑓(𝑥) = 2
𝑥 +5𝑥−6
5. 𝑓(𝑥) = 4𝑥 2 + 𝑥 − 8
3. 𝑓(𝑥) = √2𝑥 + 9
6. 𝑓(𝑥) =
𝑥
√𝑥 2 −9
Generalizations:
Function Type
Linear Functions
Cubic Functions
Cube Root Functions
Rational Functions
Quadratic Functions
Square Root Functions
Absolute Value Functions
Domain
Range
*We can also determine the zeros of a function based on the equation, rather than the graph!
THINK: If the zeros of a function are simple the x-value of the x-intercept, how could we determine the
zeros algebraically?
Examples: Find the zero(s) of each function. If no zero exists, write “NONE.”
1. 𝑓(𝑥) = −4𝑥 + 7
2. 𝑓(𝑥) = 𝑥 2 − 6
4. 𝑓(𝑥) = |𝑥| + 10
3. 𝑓(𝑥) = √−8 + 𝑥
5. 𝑓(𝑥) = 𝑥 2 + 𝑥 − 12
*To determine the range “algebraically,” think of the shape of the graph!
When in doubt, sketch it out!! 
Examples: Give the domain, range and zero(s) of each function.
1.
𝑓(𝑥) = 3𝑥 − 9
2. 𝑔(𝑥) = √𝑥 + 1
Domain:
Domain:
Range:
Range:
Zero(s):
Zero(s):
3. 𝑓(𝑥) = |𝑥| − 8
4. 𝑔(𝑥) = 𝑥 2 − 4
Domain:
Domain:
Range:
Range:
Zero(s):
Zero(s):
HOMEWORK: pg 122 WE #1,2, 9-12abc