Relation
A rule that associates a value in the domain with a value in the range.
Function
A function is a relation (rule) that assigns each element in the domain to
exactly one element in the range.
Domain
The set of all the values which may be input into
Decomposition of functions is the reverse of composition of functions. Instead
of combining two functions to get a new function, you're breaking apart a
combined function into its separate components. There is often more than one
way to decompose a functi
Equivalent Operations
The following operations can be used to generate an equivalent equation
Remove grouping symbols
Combine like terms
Reduce fractions
Add the same quantity to both sides of an equation
Subtract the same quantity from both sides of
Imaginary number, i
The square of no real number can be negative. That, however, leaves certain
equations (like x2+1=0) with no real solution. We mathematicians don't like
things not to have an answer. So, we came up an imaginary number i such
that i2 = -
Part of the beauty of mathematics is that almost everything builds upon
something else, and if you can understand the foundations, then you can
apply new elements to old. It is this ability which makes comprehension of
mathematics possible. If you were to
Arithmetic Combinations of Functions
The sum, difference, product, or quotient of functions can be found easily.
Sum
(f + g)(x) = f(x) + g(x)
Difference
(f - g)(x) = f(x) - g(x)
Product
(f g)(x) = f(x) g(x)
Quotient
(f / g)(x) = f(x) / g(x), as long as g(
Slope of a Line
The slope is represented by the letter m.
The slope of a non-vertical line is defined several ways. It is the rise over the
run. It is the change in y over the change in x.
For two points (x1,y1) and (x2,y2) where x1 x2, the slope is m = (
One-to-One Functions are wonderful things.
When solving equations, you can add the same thing to both sides, subtract
the same thing from both sides, multiply both sides by the same non-zero
thing, and divide both sides by the same non-zero thing and stil
Inverses
A function normally tells you what y is if you know what x is. The inverse of a
function will tell you what x had to be to get that value of y.
A function f -1 is the inverse of f if
-1
for every x in the domain of f, f [f(x)] = x, and
-1
-1
fo
Intercepts
x-intercept
The x-intercept is the point where the graph of an equation crosses the
x-axis. The x-intercept can be found by substituting y=0 into the
equation and solving for x. The x-intercept is also called a solution, root,
or zero of the eq
Graph of a function
The graph of a function f is the set of all ordered pairs ( x, f(x) ) where x
is in the domain of f.
Increasing Function
A function is increasing on an open interval if the function rises (positive
slope) on the interval as you move fr