Page 161 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Introduction

Functions are mathematical ideas that take one or more variables and produce a variable. You
can think of a function as a cook that takes one or more ingredients and cooks them up to make
a dish. Depending on what you put in, you can get very different things out. Moreover, not all
functions are the same. If you give one cook peanut butter, jelly, and bread, he may make a
sandwich, whereas another cook may start to sculpt a volcano with the peanut butter, and use
the jelly for lava after discarding the bread.

6.1 Functions

In an abstract mathematical sense, a function is a mapping of some domain onto some range. For
each item in the domain, there is a corresponding item in the range of the function. Thus, the
domain is all of the possible inputs to the function and the range is all of the possible outputs.
Each item in the domain corresponds to a specific item in the range. However, an item in the
range may correspond to multiple items in the domain.

For example, let’s describe a function for album titles. Our function will take as its domain,
album titles. Our function, let’s call it FL (album title) will output the first letter of the first word
in the title of the album. Thus, the range of our function will be all of the inputs.
For most of Algebra, functions are described as things that take a number and put out a
1
number. In higher mathematics, this is described as R 1 R . This means that the real
1
number line (R ) is being mapped to the real number line. If however, we have two inputs and
2
2
1
one output, we have a function that is described as R R , or the real plane(R ) is being
mapped to the real number line. Generally, we can have a function described by any
R R .
M
N
Let’s start with an old favorite the line.
f(x) = 2*x
Here, f is a function that is defined to take one variable x. It takes that one variable and doubles
it. We can plot this graph on a Cartesian grid by taking x along one axis and f(x) along the other.
Because f(x) is simply a constant, that is the number 2, multiplied by x, we know that f(x) is a line.
Assuming that we are totally ignorant, let us proceed as though we know nothing at all. To draw
a function that is new to us, here is what we normally will do (at least to begin with): We will
construct a Table 6.1. In one column, we will list various values for x that we would like to try to
see what comes out. In the other column, we will list the values of f that we get when we stuff our
values into the function. Next, on a piece of grid paper, we will plot the points, going over on the
x axis to the number we chose for x, and on the y axis to what we got out for f(x). Finally, we
will connect the dots for a rough view of what our function looks like. (More complex functions
need lots of dots!) For f(x) = 2*x, here’s what we get:

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