1. Warm-up (10 min.)
I. Factor the following expressions completely over the real numbers.
3x3 – 15x2 + 18x
x4 + x2 – 20
II. Solve algebraically and graphically
x(2x-1) = 10
x + x = 1 _

2. 1.2 Functions and Their Properties
Today you will be able to:
Represent functions numerically, algebraically, and graphically
Determine the domain and range for functions
Analyze function characteristics such as extreme values, symmetry, asymptotes, and end behavior.

3. Parts of a Function Variables Dependent Variables
These are the quantities in a function that can change
Dependent Variables
Depends on another quantity
Independent Variables
These quantities tend to have an affect on the dependent variable.

4. Parts of a Function - Example
Variables
Time and temperature
Dependent Variables
Temperature (depends on the time of day)
Independent Variables
Time (is unaffected by the temperature)
Relating Variables
Often written as a ordered pair with the independent variable first
(time, temperature)

5. Notation of a Function
In general we use x to represent the independent variable and y to represent the dependent variable.
We write…
y = f(x) meaning y is a function of x
T = f(t) meaning Temperature is a function of time

6. A Function Box
We put in an input (independent variable) apply a function and receive an output (dependent variable).
FUNCTION
Input
Independent variable
Output
Dependent variable
f(x) = x + 2 4 6

7. Domain and Range Setting up our graph
While the coordinate plane continues infinitely in all directions, most data will not.
Therefore we must determine a logical domain for the independent data.
Domain: the set of logical values for the independent variable
This will affect the range of values that make-up the dependent variable
Range: the set of values for the dependent variable that correspond to specific independent variables

8. Mappings
Uses directed arrows to connect elements of the domain to the elements of the range 5 10 15 20 3 5 5 10 15 20 3 5
Which mapping represents a function? Explain

9. Ex2 Does x = 2y2 define y as a function of x?
Vertical Line Test
A graph (set of points (x,y)) in the xy-plane defines y as a function of x if and only if no vertical line intersects the graph in more than one point.

10. What is your domain? What is the domain of y = x2?
What is the domain of A = s2?
Is there a difference between these two questions?
Could there be a difference?
Implied domain – the domain of a function defined by an algebraic expression
Relevant domain – the domain that fits a given situation

11. Ex 3 Pair with a neighbor to find the domain of each function.
, where A(s) is the area of an equilateral triangle with side of length s.

12. Finding Range
Ex 4 Find the range of f(x) = 10 – x2.

13. Graphs of functions in reverse
Ex 5 Sketch the graph of a function that has domain [-5,-1]  (2,4] and range [-4,-1]  [1,).

14. Continuity
Important property of the majority of functions that model real-world behavior
Graphically speaking, continuity investigates at a point whether a function comes apart at that point.
Continuous at all x = a Discontinuous at x = a a

15. Removable or Non-removable? That is the question!!
Discontinuities can be removable or non-removable
Removable discontinuity
Removable discontinuity a
Non-removable discontinuities a a a
Jump discontinuity
Infinite discontinuity

16. Ex 6 Graph each of the following
Ex 6 Graph each of the following. Which of the following are discontinuous at x = 2? Are any of the discontinuities removable?
a) f(x) = x + 3 x – 2 b) g(x) = (x + 3)(x – 2) c) h(x) = x2 – 4

17. Ex 7 Label each graph (or parts) as increasing, decreasing, or constant.1 1 1 1 1

18. Increasing, Decreasing, and Constant Function on an Interval
A function f is increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in f(x).
A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x).
A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x).

19. Ex 8 Tell the intervals on which each function is increasing and the intervals on which it is decreasing.
f(x) = (x - 2)2 g(x) = x2 x2 - 1

20. Boundedness
A function f is bounded below if there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f.
A function f is bound above if there is some number B that is greater than or equal to every number in the range of f. Any such number B is called an upper bound of f.
A function f is bounded it is bounded both above and below.

21. Ex 9 Identify each of these functions as bounded below, bounded above, or bounded.
w(x) = -3x2 + 4 p(x) = x x2+1

22. Warm-Up (8 min.)
Determine the domain and range of each function. 1. y = 1+ x 2. f(x) = x x State the intervals on which each graph above is increasing or decreasing.

23. Local and Absolute Extrema
A local maximum of a function f is a value f(x) that is greater than or equal to all range values of f on some open interval containing c. If f(x) is greater than or equal to all range values of f, then f(c) is the maximum (or absolute maximum) value of f.
A local minimum of a function f is a value f(x) that is less than or equal to all range values of f on some open interval containing c. If f(x) is less than or equal to all range values of f, then f(c) is the minimum (or absolute minimum) value of f.
Local extrema are also called relative extrema.

24. Ex 10 Decide whether f(x) = x3 – 2x2 + 5x – 2 has any local extrema.

25. Symmetry
A graph appears identical when viewed on either side of a line or through a point.
Symmetry can be easily viewed graphically, numerically, or algebraically.

26. Summarizing symmetry Type of Symmetry Graphically Numerically
Algebraically
With respect to the y-axis
f(x) = x2
f(-x) = f(x) called even functions
With respect to the x-axis
x = y2
*Not a function but (x,-y) is on the graph whenever (x,y) is on the graph
With respect to the origin
f(x) = x3
f(-x) = - f(x) called odd functions

27. Ex 11 Tell whether each of the following functions is even, odd, or neither.
f(x) = x2 – 3
g(x) = x2 – 2x – 2
h(x) = x3
4 – x2

28. Asymptotes
The line y = b is a horizontal asymptote of the graph of a function y = f(x) if f(x) approaches a limit of b as x approaches +∞ or - ∞.
In limit notation,
The line x = a is a vertical asymptote of the graph of a function y = f(x) is f(x) approaches a limit of + ∞ or ∞ as x approaches a from either direction.

29. Ex 12 Identify any vertical or horizontal asymptotes by viewing each graph and observing its end behavior.

30. Tonight’s Assignment
P Ex 3-66 m. of 3