where a_{0}, a_{1}, a_{2},...a_{n} are constants and n is a nonnegative integer. n denotes
the "degree" of the polynomial.

You should be familiar with the common names of certain polynomial functions. A
second-degree polynomial function is a quadratic function
(f (x) = ax^{2} + bx + c). A first-degree polynomial function is a linear
function (f (x) = ax + b). Finally, a zero-degree polynomial function is a simply a
constant function (f (x) = c).

Rational Functions

A rational function is a function r of the form

r(x) =

where
f (x) and g(x) are both polynomial functions. For example,

r(x) =

is a rational function. Note that we must exclude from the
domain of r(x) any value of x that would make the denominator, g(x) equal zero,
since this would make r(x) undefined. Thus, x = 0 is not in the domain of the
function r(x) we just defined above.

Even and Odd Functions

Another useful classification of functions is even and odd.
For an even function, f (- x) = f (x) for all x in the domain.
This sort of function is symmetric with respect to the y-axis. For example:

For an odd function, f (- x) = - f (x) for all x in the domain.
This sort of function is symmetric with respect to the origin. For example:

Composite Functions

As we know, f is a function that can take an input x and transform it into an output
f (x). Similarly, f can take the output of another function, such as g(x) as
its input, and transform that input into f (g(x)). When two functions are combined so
that the output of one function becomes the input for the other, the resulting combined
function is called a composite function.
The notation for the composite function f (g(x)) is (fog)(x).

Example:

If f (x) = 3x + 4 and g(x) = 2x - 7, then how could we find (fog)(2)?

Solution:

The problem is asking us to find f (g(2)). One way is to work step-by-step with g and then with f:

g(2)
= 2(2) - 7
= -3

Now we use g(2) = - 3 as the input for f:

f (g(2))
= f (- 3)
= 3(- 3) + 4
= -5

A second way would be to solve for (fog)(x)
directly.

Now, we can plug x = 2 into this function: f (g(2)) = 6(2) - 17 = - 5

Piecewise-Defined Functions

One type of function we'll be dealing with often in calculus is the
piecewise-defined function. These functions are defined differently for different intervals in
their domain. For example, consider the following piecewise function:

f (x) =

For x less than or equal to 2, f (x) is defined by f (x) = x^{2}. For x greater than 2,
f (x) is defined by f (x) = 2x. Thus, f (1) = 1^{2} = 1, and f (4) = 2(4) = 8. The graph of
this function is below:

Interval Notation

Finally, we should briefly mention interval notation, which we'll be using
throughout the rest of the guide. An interval is a set of all numbers between two
endpoints. An closed interval includes both of the endpoints, while an
open interval includes neither of the endpoints.
So,
[a, b] means the set of all x such that a≤x≤b
(closed interval)
(a, b) means the set of all x such that a < x < b (open interval)
Intervals can also be half-open (and half-closed). For example, [a, b) is closed at x = a
and open at x = b. This interval represents
a≤x < b
Intervals that have infinity as an endpoint should always be open at infinity, since no
interval can actually contain infinity. Thus, "all numbers less than 4" should be
written as
(- ∞, 4]
, while "the set of all real numbers" should be written as
(- ∞,∞).