Intro to Graphs
2 is a real number, and 3 is a real number. We can take those two numbers
and write them as a pair of real numbers: (2, 3). When we write a pair of
real numbers, the order is important. That is to say that (2, 3) is not the
same pair as (3
Inverse Functions
Inverse Functions
One-to-one
One-to-one
Suppose f A B is a function. We call f one-to-one if every distinct
f A
B a function. We call f
Suppose f : A B is a function. We call f one-to-one if every distinct
pair of objects in A is assigne
Constant & Linear Polynomials
Constant polynomials
A constant polynomial is the same thing as a constant function. That is, a
constant polynomial is a function of the form
p(x) = c
for some number c. For example, p(x) = 5 or q(x) = 7.
3
The output of a co
More on functions
Suppose f : R R is the function dened by f (x) = x5 . The letter x in
the previous equation is just a placeholder. You are allowed to replace the x
with any number, symbol, or combination of symbols that you like.
f (4) = 45
f (1) = (1)5
n-th Roots
Cube roots
Suppose g : R R is the cubing function g(x) = x3 .
We saw in the previous chapter that g is one-to-one and onto. Therefore,
g has an inverse function.
The inverse of g is named the cube root, and its written as 3 . In other
words, g
Quadratic Polynomials
If a > 0 then the graph of ax2 is obtained by starting with the graph of x2 ,
and then stretching or shrinking vertically by a.
If a < 0 then the graph of ax2 is obtained by starting with the graph of x2 ,
then ipping it over the x-a
Graph Transformations
There are many times when youll know very well what the graph of a
particular function looks like, and youll want to know what the graph of a
very similar function looks like. In this chapter, well discuss some ways to
draw graphs in
Roots & Factors
Roots of a polynomial
A root of a polynomial p(x) is a number R such that p() = 0.
Examples.
3 is a root of the polynomial p(x) = 2x 6 because
p(3) = 2(3) 6 = 6 6 = 0
1 is a root of the polynomial q(x) = 15x2 7x 8 since
q(1) = 15(1)2 7(1
Division
We saw in the last chapter that if you add two polynomials, the result is
a polynomial. If you subtract two polynomials, you get a polynomial. And
the product of two polynomials is a polynomial.
Division doesnt work as well. Sometimes when we div
Basics of Polynomials
A polynomial is what we call any function that is dened by an equation
of the form
p(x) = an xn + an1 xn1 + + a1 x + a0
where an , an1 , .a1 , a0 R.
Examples. The following three functions are examples of polynomials.
p(x) = 2x2 x 2