1. Lecture 2: Properties of Functions

2. Domain & Range
Domain:
The domain of a function is the set of all the possible values of the independent variable (all the possible x-values) of your function.

3. Domain & Range
e.g. Domain : [-3 , 4]

4. Domain & Range
Domain: [-4, -2] U [1,5]

5. Domain & Range
Range:
The range of a function is the set of all the possible values of the dependent variable (all the possible y-values) of your function.

6. Domain & Range
e.g.
Range: [-5 , 3]

7. Domain & Range
e.g.
]5,11] U
[0,4]
Range: [0 , 4] U ]5, 11]

8. Variations “Variations” describe what is happening to function:
i.e. Under which domain intervals does the function increase, decrease, or stay constant.
NOTE: The way I described this in class last week was very different from the description the book gives so please pay attention. I made a mistake apparently.

9. Variations
A function is increasing if it is represented by a line or curve that is ascending (going up) from left to right or constant!!!.
e.g.
Increasing: [-3, 3]

10. Variations
A function is decreasing if it is represented by a line or curve that is descending (going down) from left to right or constant!!!.
e.g.
Decreasing: [-2, 6]

11. Variations
A function is constant if it is represented by a horizontal line (going straight) from left to right.
e.g.
Constant: [-2, 3]

12. Extrema: Minimum & Maximum
The Minimum of a function is the smallest value of the dependent variable. (Smallest y-value)
The Maximum of a function is the largest value of the dependent variable. (Largest y-value)

13. Extrema: Minimum & Maximum
e.g.
Minimum: y = Maximum: y = 5

15. Sign of the Function: +’ve and –’ve
Using x-intervals…..
A function is Positive if the values of the dependent variable are positive.
A function is Negative if the values of the dependent variable are negative.

16. Sign of the Function: +’ve and –’ve
e.g.
Positive: [ -5 , -3 ] U [ 4 , 6]
Negative: [-3 , 4 ]

17. Sign of the Function: +’ve and –’ve
The Zero(s) of a function are the x-values where the function crosses the x-axis.
Zeros: x = -3
&
x = 4

18. Sign of the Function: +’ve and –’ve
The Initial Value of a function are the x-values where the function crosses the y-axis.
Initial value: y = -3