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. Homework On HW Worksheet Page of Website 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. On HW Worksheet Page of Website
Homework
On HW Worksheet Page of Website
Page 27:
28, 29, 30
All Parts