### Help: Miscellaneous functions

Here is a complete list of miscellaneous functions accepted by QuickMath.
The tables show the usual form in which the functions appear in textbooks,
along with the form accepted by QuickMath. In most cases, the QuickMath
version is identical to the textbook version.

If there is a function missing which you would like see added to those
supported by QuickMath, just send your suggestion to contact form.

Functions overview

The concept of a function is central to mathematics. As we shall see,
functions are used to model special relationships between quantities and an
understanding of how functions work and different ways to represent functions
will be invaluable for solving a variety of problems. In particular, the ability
to analyze a function graphically is a significant benefit when the function is
modeling a relationship between two quantities that is nonlinear, so that rate
of change is not constant.

## Functions from Numerical and Algebraic View

**OBJECTIVES**

1. Define the Concept of Function

2. Describe Functions Numerically and Algebraically

3. Use Functional Notation

2.1 Functions from a Numerical and Algebraic View

Consider the following statements.

- "Income tax is a function of how much you earn."
- "My grade on an exam is a function of how much time I study."
- "TV ad price is a function of ratings."

We could easily replace the phrase "is a function of' with "depends on" in
each statement. This kind of relationship between quantities, where one
quantity depends on another quantity (or quantities), is a fundamental idea
in mathematics. We use the term

*function* to refer to any such
relationship. In particular, we will investigate the idea of one quantity
being a function of another quantity.

**What Is a Function?**

As just stated, in the most general case we think of a function as a
relationship between quantities, where one quantity determines the other. It
is useful, when we say "quantity B is a function of quantity A," to think of
particular values of quantity A as inputs and corresponding values of
quantity B as outputs, with some rule (or process) prescribing how any
particular value of quantity A determines precisely one corresponding value
of quantity B. More formally, we define a real-valued function of a real
variable in three parts, as follows:

A function consists of:

- A collection of real number inputs, called the domain of the function
- A collection of corresponding real number outputs, called the range of
the function
- A rule of correspondence that describes how each particular input
generates exactly one corresponding output

It is important to note here that each input for a given function generates
only one corresponding output, but distinct inputs may actually generate the
same output.The following table is adapted from the 2011 Form 1040 Federal
Income Tax booklet. It describes federal income tax owed as a function of
taxable income for a filer filing as single. It shows, in theory, how every
taxable income level generates a corresponding tax liability.

The domain of the "income tax function" is al1 possible taxable incomes,
represented by the real number interval (0, oo). The range is the collection of
all possible corresponding tax amounts, which the interval (0, oo) also
represents. A question of some merit is, "How do we describe functions?" In some
cases, a verbal description of domain, range and rule of correspondence may
indeed be pos sible (imagine doing this with the income tax function above), but
it is typically not a practical approach. In the situation above, the table
provided an adequate display of the rule of correspondence between taxable
income and tax liability, and from the context we were able to deduce a domain
and range. We now tum our attention to ways of describing functions.

**Describing Functions Numerically**

Table 2 has as its basis information from the periodical The New York Times.
It lists the approximate U.S. oil imports from Mexico for 2001-2006.

This table describes oil imports,I, as a function of year, t. The domain is
the set D = { 200 I, 2002, 2003, 2004, 2005, 2006} while the range is the set R
= { l.35, 1.5, 1.55, 1.6)

The table gives the rule of correspondence between t and *I*. Notice,
however, that the table does not describe t as a function of *I*, as the
value I = 1.5 corresponds to t = 2002, t = 2005, and t = 2006. We say here that
I is described numerically as a function of t, as the table gives specific
values of I corresponding to specific values of t.

In the situation given above, since the description of I was as a function of t,
we say that t is the independent variable and I is the corresponding dependent
variable. The distinction between independent variables and dependent variables
will become more important later.

Table 3 below gives the median age of the U.S. population for certain years, as
recorded by the U.S. Bureau of the Census in Statistical Abstract of the United
States, 1998. The table describes median age as a function of year.

We will use the notation M(y) (read "M of y") to denote the median age of the
U.S. population in year y. Specifically, M( 1920) is the median age in 1920,
M(1930) is the median age in 1930, and so on. Consequently, we write M(1920) =
25.3, M(1930) = 26.4, etc. We call this notation functional notation, and it is
a useful shorthand for describing the relationship between inputs and outputs
for a given function. In particular, mathematicians commonly use it when
describing functions algebraically. We now turn our attention to that method of
describing functions.

**Describing Functions Algebraically**

When we use an explicit formula or an equation to describe how inputs and
outputs are related, we say that we are describing a function algebraically. For
instance, consider the familiar formula for the area, A, of a circle of radius
r: A = pi*r^{2}. Considering r as the independent variable, this
equation describes A as a function of r. For each particular radius, we have an
explicit set of instructions for how to calculate the corresponding area of the
circle with that radius, namely, "square the radius

and multiply the result by x" In this context, a natural domain and a
corresponding range arise for the function in question. The domain is all
possible radii r (so, all positive real numbers), and the range is the set of
all corresponding areas, which is also the set of all positive real numbers.
Using interval notation, the interval (0, oo) gives both the domain and the
range.

The usefulness of functional notation becomes more apparent when working with
functions described algebraically. For instance, consider the equation f(x) = 3x^{2} + 4. As before, when using this notation, x represents the input
and the symbol f (x) represents the corresponding output. The letter f is
nothing more than a "nick name" for the function described. The equation, taken
as a whole, gives the rule of the function: