1. Real-Valued Functions
A Real-Valued Function f of a real variable x from set X to set Y is a correspondence that assigns to each number x in set X exactly one number y in set Y X
Domain x Y f
Range
Y = f(x)
X and Y are sets of real numbers
This sounds familiar but the words seem different.
The domain of f is the set X. The number y is the image of x under f and is denoted by f (x). The range of f is a subset of Y and consists of all images of numbers in set X

2. Implicit vs Explicit Form
Defines y, the dependent variable, as a function of x, the independent variable.
Implicit Form
Defines the equation in terms of y. Use the Explicit Form to evaluate the function.
Explicit Form
f(x) and y are interchangeable.
Function Notation

3. Evaluating Functions
Evaluate each of the following.

4. Evaluating Functions
Evaluate each of the following.

5. More Evaluating Functions
Evaluate
for
This looks hard, but it really isn’t that bad.

6. Evaluating Functions Worksheet
Homework
Evaluating Functions Worksheet

7. Homework Problem 1 Asi de Facil Evaluate for
Holy Schnikies! That’s really pretty darn easy.
Asi de Facil

8. Homework Problem 2
Evaluate
for
That was easy

9. Homework Problem 3
Evaluate
for
That was easy

10. Evaluate
for

11. Homework
Evaluating Functions
Worksheet #2

12. Intervals on the Real Line
Set Notation
Interval Notation
Graph
( )
Bounded Open Interval
(a, b)
a b x
[ ]
Bounded Closed Interval
[a, b]
a b x
[ )
[a, b)
Bounded Intervals (neither open or closed)
a b x
( ]
(a, b]
a b x )
Unbounded Open Intervals
a b x (
a b x ]
Unbounded Closed Intervals
a b x [ x
a b
Entire Real Line
a b x

13. Domain and Range of a Function
The domain of a function can be described implicitly, or the domain can be described explicitly.
This equation has an implicitly defined domain that is the set
This equation has an explicitly defined domain given by
An implied domain is the set of all real numbers for which the equation is defined.
An explicitly defined domain is one that is given along with the function.

14. Finding the Domain and Range of a Function
Find the domain and range of
The domain is the set of all x-values for which
That means the domain is the interval
Since f(x) is never negative, that means the range is the interval
Let’s look at a diagram.

15. More Domain and Range of a Function
Find the domain and range of each of the following functions.
If both polynomials are the same degree, divide the coefficients of the highest degree terms.
If the numerator is lower degree than the denominator, y = 0 is the asymptote.
There is a horizontal asymptote at y = 1
If the numerator is higher degree than the denominator, there is no asymptote.
Asi de Facil

16. Find Domain & Range for all functions
Homework
Page 27:
13 – 16, 19, 20, 25, 26
Find Domain & Range for all functions

17. Functions Defined by More than One Equation
Find the domain and range of
Let’s look at a diagram.
This is called a Piecewise Function.
That was easy

18. Piecewise Functions 1 That was easy
Evaluate the function as indicated. Determine the domain and range.
That was easy

19. Piecewise Functions b That was easy
Evaluate the function as indicated. Determine the domain and range.
That was easy

20. Piecewise Functions 3 That was easy
Evaluate the function as indicated. Determine the domain and range.
That was easy

21. Piecewise Functions d That was easy
Evaluate the function as indicated. Determine the domain and range.
That was easy

22. Piecewise Functions 5 That was easy
Evaluate the function as indicated. Determine the domain and range.
That was easy

23. Homework
Page 27:
28, 29, 30
All Parts

24. Graphs of Functions
The graph of a function y = f(x) consists of all points (x, f(x) ), where x is in the domain of f.
x = the directed distance from the y-axis.
f(x) = the directed distance from the x-axis.
f(x) x

25. Graphs of Basic Functions
Identity Function
Squaring Function
Cubing Function
Square Root Function

26. More Graphs of Basic Functions
Absolute Value Function
Rational Function
Sine Function
Cosine Function

27. Vertical Line Test
A graph represents a function if any vertical line drawn intersects it in at most one point.
The vertical line never intersects the graph in more than one point, therefore this is a function.
The vertical line never intersects the graph in more than one point, therefore this is a function.
The vertical line does intersect the graph in more than one point, therefore this is not a function.

28. Transformations of Functions
Let’s take a look at the graph of
Original function
Horizontal shift c units to the right
Horizontal shift c units to the left
Vertical shift c units downward
Vertical shift c units upward
Reflection (about the x-axis)
Reflection (about the y-axis)
Reflection (about the origin)

29. Transformation Example
Use the graph of f shown in the figure to sketch the graph of each function.

30. Homework
Page 28:
41 – 44, 55, 56

31. I didn’t realize that this was that easy.
Polynomial Functions
n is the degree of the function, the numbers ai are the coefficients
an is the leading coefficient, and a0 is the constant term
Let’s look at a sample with some actual numbers.
4 is the degree of the function,
3 is the leading coefficient,
and -7 is the constant term
5is the degree of the function,
-6 is the leading coefficient,
and 3 is the constant term
I didn’t realize that this was that easy.
That was easy

32. Turning Points of Functions
The number of turning points that a polynomial function has can be determined by the degree of the equation.
The number of turning points of a polynomial function is equal to n - 1
One turning point
Two turning points
Four turning points

33. End Behavior of Functions
There are two factors that determine the End Behavior of a function.
End Behavior of Functions
Degree of the Function
Even
Polynomials of even degree have ends going in the same direction.
Polynomials of odd degree have ends going in opposite directions.
Odd
Sign of the Leading Coefficient
Even
Odd
Even
Odd
Up to left
Up to right
Down to left
Up to right
Down to left
Down to right
Up to left
Down to right
That’s a little confusing but pretty easy.

34. Describing the Behavior of a Function
For each of the following functions
a) state the number of turning points b) describe the end behavior
Positive
Even
3 turning points
Up to left
Up to right
Negative
Odd
Up to left
Down to right
4 turning points
Negative
Even
Negative
Even
5 turning points
Down to left
Down to right
Asi de Facil
3 turning points
Down to left
Down to right

35. Function Behavior Worksheet
Homework
Function Behavior Worksheet

36. Zeros of a Function
The zeros of a function are the solutions of the equation f(x)=0
Let’s take a look at some examples
Example 1
Example b
Since f(0) = 0, f(-1) = 0
and f(1) = 0 the zeros of the function are:
x = 0, x = -1, x = 1
Since f(4) = 0, x = 4 is a zero of the function.

37. Even and Odd Functions
An Even Function is symmetric with respect to the y-axis.
Test for Even and Odd Functions
An Odd Function is symmetric with respect to the origin.
Example 1
Example b
Example 3
The Function is Even
The Function is Odd
The Function is Odd

38. More Even and Odd Functions
Find the coordinates of a second point on the graph of a function if the given point is on the graph and the function is (a) even and (b) odd.
That was easy
Remember:
Example 1
Example b
Example 3
(-1, -4)
(3, 7)
(-6, 2)
(1, -4)
even
(-3, 7)
(6, 2)
(1, 4)
odd
(-3, -7)
(6, -2)

39. Practice Examples for Even and Odd Functions
Example b
Example 3
Note:
Even
Even
Example d
Odd
Note:
Even

40. Homework
Page 29: