1. Functions
Lesson 1.3

2. Today we look at the mathematical way of talking about the "rule"
The Magic Box 1 -3 4 3 3 1
Consider a box that receives numbers in the top
And alters them somehow and sends a (usually) different number out the spout
Today we look at the mathematical way of talking about the "rule"
For each of the number combinations, can you figure out the "rule" which alters the number?

3. What is a Function? Definition: A function is a rule
Given X = { x1, x2, …} and Y = {y1, y2, …}
Assign to each element of X a unique element of Y
The set X is the domain
the set of all possible x values
The set Y is the range
the set of all resulting y values

4. Notation y = f(x) can be thought of as ..
“y is the image of the function f at x” or
“y is the value of the function f at the point x”
It is often read “y equals f of x”
your instructor will often use the phrase “f at x …”
this reflects the above

5. Alternate Definition
A function is a set of ordered pairs: f(x) = { (x1, y1), (x2, y2), … }
Where no two ordered pairs have the same first element
Which of these is a function?
{ (3,4), (6,0), (9,4), (7,2)}
{(1, 2), (3, 4), (5, 6), (1, 7)}
{(8,-1), (9, -1), (10, -1), (105, -1)}

6. The -> is the STO> key
Function Notation
Normal notation:
Functions can be defined/declared in your calculator:
Note the use of functional notation f(3) to evaluate functions.
The -> is the STO> key

7. Piecewise Functions
Some functions may be defined differently for different portions of the domain.
Your calculator can also define piecewise functions

8. Piecewise Functions Note the results when the function is graphed:
Actually there should be a gap between 6 and the square root of 5
Which of the above values should apply for the function
Try the F3, trace on the graph

9. Domain and Range Either the domain or range or both can be restricted
due to the nature of the function
Consider
Determine the domain and range for each:

10. Domain and Range f(x) g(x) Domain: all real numbers
BUT … not zero … why?
Range: y < -2 or y > 2
g(x)
Domain: x < -3 or x > -2 (why?)
Range: y >= 0

11. Composition of Functions
Basic Concept:
value fed to first function
result fed to second function
end result taken from second function
Oft used notation: y = g(f(x)) or

12. Composition of Functions
Example – given :
Try these f(4) = ? g(f(4)) = ? f(g(-2)) = ?
Try defining the functions on your calculator and using the notation !!

13. Assignment Lesson 1.3A Page 31
Exercises: 3, 7, 11, 15, 21, 25, 27, 37, 43, 49

14. Defining a Graph The graph of a function is:
the set of all points (x,y) which …
satisfy the function y = f(x)

15. Some Graphs are NOT Functions
Which are not functions?
Use the “vertical line” test.

16. Intercepts
We are often concerned with where the graph intersects the axes
x-intercepts => when f(x) = y = 0
y-intercepts => when x = 0, f(0)

17. Reference Table of Functions
Page 27, Text
Linear
Quadratics
Cubic
Absolute
Root functions
Reciprocals
Trig functions

18. Symmetry of Graphs Symmetric with the y-axis f(-x) = f(x) for all x
Called even functions

19. Symmetry of Graphs Symmetric about the origin f(-x) = -f(x)
Called odd functions
Note: There are many functions which are neither odd nor even

20. Transformation Shift a function up or down y – k = f(x)
k > 0 => up
k < 0 => down
Shift function right or left y = f(x – h)
h > 0 => right
h < 0 => left

21. Try for the following function:
Transformation
Define the function on the home screen
Enter different versions of f(x + h) f(x)+k a*f(x) f(-x) -f(x)
Try for the following function:

22. Assignment
Lesson 1.3B
Pg 31
Exercises 51, 53, 57, 59, 63, 67, 69