Function Algebra and Important Functions
Function notation
We write f(x) to mean the function whose input is x.
Examples:
If
f(x) = 2x - 3
then
f(4) = 2(4) - 3 = 5
We can think of f and the function that takes the input multiplies it by 2 and subtracts 3.
Relations and Functions
Relations
A relation is a rule that takes an input from a set (called the domain) and gives one or more
outputs of another set (called the range).
Examples
1.
(0,0), (4,4), (0,3), (2,1)
2.
y = 2x
3.
4. The circle is a relation. We
Inverses
Inverses
Definition
The inverse of a relation R is the relation consisting of all ordered pairs
(y,x) such that (x,y) belongs to R
Example:
The inverse of the relation
(2,3), (4,5), (2,6), (4,6)
is
(3,2), (5,4), (6,2), (6,4)
Generally we switch t
Completing the Square and The Square Root Method
The Square Root Property
If we have
x2 = k
then we can take the square root of both sides to solve for x.
The Square Root Property
For any positive number k,
if
x2 = k
then
x =
or
x = -
Example
Solve
x2 - 6
Decimal Review
Fill in the blanks:
1. We use the word
only to indicate the separation between the whole number part and the
fractional part.
2. To change a decimal fraction to a decimal number, make sure the number of digits to the right of the
decimal po
Multiplication of Real Numbers
I. Some Definitions
When multiplying two numbers such as
2x3 = 6
The numbers on the left (the 2 and the 3) are called factors and the result (the 6) is called the
product.
II. Properties of Multiplication
A. The Zero Propert
Parabolas
Parabolas
Consider the plot of
y = x2
to the right
The point at the bottom (or top) is called the vertex and the line that
cuts the parabola into two equal pieces is called the axis of symmetry.
Example
sketch the graph of the parabola
y = x2 -
Special Functions
Square Root Functions
You can identify a square root function by the equation
f(x) =
They are half parabolas on their sides.
Absolute Value functions
You can identify an absolute value function by the equation
y = |x|
These look like the
Understanding Whole Numbers and Addition
I. The Number System and Base 10
The number system is constructed based on the principal that we have ten fingers. When we count
after 9 we arrive at the two digit number 10. Similarly after 99 we arrive at a three
The Parabola
Algebraic Definition of The Parabola
Recall that the standard equation of the parabola is given by
y = a(x - h)2 + k
If we are given the equation of a parabola
y = ax2 +bx + c
we can complete the square to get the parabola in standard form.
G
Subtracting Whole Numbers
I. No Borrowing
To subtract whole numbers we write them as in an addition problem and subtract each digit
moving from the right to the left.
Example:
789
- 34
755
Note that 9 - 4 = 5,
8 - 3 = 5, and 7 - 0 = 7
II. Borrowing is Nec