THE LANGUAGE OF FUNCTIONS
*
In algebra you began a process of learning a basic vocabulary, grammar, and syntax
forming the foundation of the language of mathematics. For example, consider the
following algebraic sentence:
Solve for
x
if
8
5
3
2
=
+
−
x
x
You learned to read this sentence, understand what it means, what it is asking for, and in
addition, how to go about answering the question.
In precalculus, it is important to learn and understand the language of functions. This
includes understanding function notation and sentences using function notation. Using
the language of functions, for example, we can restate the same question above in the
following way:
If
when
does
5
3
)
(
2
+
−
=
x
x
x
f
?
8
)
(
=
x
f
OR
If
for what value(s) of
x
does
5
3
)
(
2
+
−
=
x
x
x
f
8
)
(
=
x
f
?
OR
If
and
for what value(s) of
x
does
5
3
)
(
2
+
−
=
x
x
x
f
8
)
(
=
x
g
)
(
)
(
x
g
x
f
=
?
OR
If
and
when
does
5
3
)
(
2
+
−
=
x
x
x
f
8
)
(
=
x
g
)
(
)
(
x
g
x
f
=
?
Using the language of functions, these are just a few ways to express the same question
first stated above “algebraically”. What is important to note here is that the methods for
finding the answer to the question may be the same as in algebra (and the process of
finding the answer may be easy once you understand what is being asked): the difficult
part may be understanding what is being asked.
That is why it is important for you to
understand
functions and function notation.
I.
You should understand what makes an assignment a function. You should be able to
identify if an assignment is a function whether the function is given as an assignment
using sets, tables, ordered pairs, equations, or graphs. You should be able to identify
domains (including natural domains) and ranges using sets, tables, ordered pairs,
equations, or graphs. You should also know how to (and when to) use the graphing
calculator to help answer questions about functions.
_______________________________________
*This handout is not intended as a tutorial or a complete review of functions, but rather as a quick review of
notation that we use frequently in this course. There are additional important topics such as applications,
quadratic functions, extreme values, etc., which are discussed in lecture and in the text, but are not covered
here
.
1