1. 1.2 Functions & their properties
Notes 9/28 or 10/1

2. Relations Every relation has a domain and range
Domain : x values, independent
Range: y values, dependent
Functions: x value DO NOT REPEAT
Examples: {(12, 4), (8, 3), (3, 9)}
domain: {3, 8, 12}, range: {3, 4, 9}, is a function

3. From a graph:

4. Find domain or range when given an equation:
-determine what values of x will work
Ex: 1) f(x) = x + 4
2) f(x) = x – 10
3) f(x) = 5
x - 5

5. Continuity Continuous: continuous at all values of x
Discontinuity: examples on p
- removable discontinuity: there is a “hole” in your graph
- jump discontinuity: the graph “jumps” a point(s)
- infinite discontinuity: the graph has a vertical asymptote (there is a vertical line where the graph cannot cross or touch)

6. Identify pts of discontinuity
Graph it, also see when the denominator = 0
Ex: 1)f(x) = x + 3
x – 2
2)f(x) = x2 + x – 6
3) f(x) = x2 – 4
x - 2

7. Increasing/Decreasing function
Functions can be increasing, decreasing, or constant
A function is increasing on an interval if, for any 2 pts in the interval, a positive change in x results in a positive change in f(x)
A function is decreasing on an interval if, for any 2 pts in the interval, a positive change in x results in a negative change in f(x)
A function is constant on an interval if, for any 2 pts in the interval, a positive change in x results in a 0 change in f(x)

8. Determining increasing/decreasing intervals:
look for the x values that the graph is increasing/decreasing/constant
#1 and 2 on handout
3) f(x) = 3x2 - 4

9. Boundedness
A function is bounded below if there is a minimum. Any such # b is called a lower bound of the function.
A function is bounded above if there is a maximum. Any such # B is called an upper bound of the function.
A function f is bounded if it is bounded both above and below

10. Examples of bounded
Bounded below:
Bounded above
bounded

11. Local & Absolute Exterma
Maximums/minimums – every function (w/ the exemption of a linear function)
To determine where the local maximum and/or local minimum is located look at the graph or use a calculator
Ex: #8 & 9 on handout
3) f(x) = -x2 – 4x + 5
4) f(x) = x3 – 2x + 6

12. Asymptotes
Vertical asymptotes (VA): set the denominator = 0 and solve, write answers as equations of vertical lines (x = #)
Horizontal asymptotes (HA): 3 possibilities
1) if the exponent is lower in the numerator then the denominator: the HA is y = 0
2) if the exponents are equal: the HA is y = a/b, where a is the leading coefficient in the numerator & b is the leading coefficient in the denominator
3) if the exponent is higher in the numerator than the denominator there is no HA

13. examples Identify the asymptotes: 1) f(x) = 3x x2 - 4 2) f(x)= 3x2

14. End behavior
What direction does the graph go (up or down) at the far left and far right
Ex: 1) f(x)= 3x
x2 - 1
2) f(x)= 3x2
3) f(x)= 3x3

15. Homework
Section 1.2 exercises p #2-16 even, all, even, even