1. Real Numbers In this class, we will consider only real numbers.
Intuitively, to decide if a number is a real number, ask yourself if it can be plotted on the number line.
Is it a real number: -1 ? Sqrt(-1) ?
Match each number with a point on the number line: 02, , (2/3)-1, , Pi, Sqrt(5)

2. Functions
A function is a rule that matches each input with exactly one output.
Function notation:
x is the input
f(x) is the matching output, read “f of x”
For the function, f(x) = 3x + 1, write in function notation the output for each input:
0, /3, a, a + 1

3. Domain of a Function
The set of all possible inputs for a function is called the domain of the function.
When the domain of a function is not explicitly stated, it is understood that you should include any real number in the domain if the function rule can be used to match it with a definite real number.
For f(x) = 1/x, explain why x = 0 is not in the domain. (We say f(0) is not defined.)

4. Using Limits to Describe Function Behavior
Consider the function
f(2) is not defined so 2 is not in the domain of this function.
We want to know what the trend of the outputs is when the inputs are near 2, both on the left and the right side of 2.

5. Limit concept: When the input approaches a given number in a specified way, do the outputs approach a definite number?
x is near 2 but smaller than 2, on left of 2
x is near 2 but larger than 2, on right of 2 x
f(x)
1.8
3.8
1.9
3.9
1.99
3.99
1.999
3.999
As x approaches 2 from
the left, f(x) approaches 4. x
f(x)
2.2
4.2
2.1
4.1
2.01
4.01
2.001
4.001
As x approaches 2 from
the right, f (x) approaches 4.

6. Limit Notation As x approaches 2 from the left, f(x) approaches 4.
To describe this behavior in symbols, we write
As x approaches 2 from the right, f(x) approaches 4.

7. Two-Sided Limits
If the left and right limits are the same number, then the two-sided limit exists and is that number.
As x approaches 2, the left and right limits exist and are equal for the function
so we write

8. Self-check Activity 1 Express in limit notation:
As x approaches -3 from the left, g(x) approaches -10.
2. Express in limit notation:
As x approaches -3 from the right, g(x) approaches -10.
3. Express in limit notation:
As x approaches -3, g(x) approaches -10.

9. Self-Check 1 Continued For
Set-up a table to verify that as x approaches –3 from the left, g(x) approaches –10.
Set up a table to verify that as x approaches –3 from the right, g(x) approaches –10.