Chapter 1: Functions
A function is a relation which maps each element in its domain
element in its range . For example, we write
,
rule of and is the domain of .



to one and only one
, where
is the
When writing a function, always write both its rule and domain.
2 functions are equal if and only if they have the same rule and the same domain.
Replace and
below with the actual domain and range of in interval notation.
Is a function?
Use: Vertical Line Test
Yes “Since any vertical line
, where
, cuts the graph of at one and
only one point, is a function.”
No
“Since the vertical line
, where
, does not cut the graph of at
one and only one point, is not a
function.”
Is one-one?
Horizontal Line Test
“Since any horizontal line
, where
, cuts the graph of at one and only
one point, is one-one.”
“Since the horizontal line
, where
, does not cut the graph of at one
and only one point, is not one-one.”
OR give a counter-example:
“Since
, where
,
is not one-one.”
is a restriction of if and have the same rule and
, i.e. is with its domain
restricted. Even if is not one-one, a restriction of which is one-one can be defined.

Use turning points of a function to show that it is not one-one or to restrict its domain.
Exists if and only if:
Domain
Range
(inverse function)
is one-one
(composite function)
(with
restricted to
)
Rule
Graph



1. Let
.
2. Express in terms of , such
that
.
3. Then
.
The graph of
is the graph of
reflected about the line
.
If you are required to solve
, solve
or
instead. Since
the graphs of and
must intersect on the line
, these equations are equivalent.
The rules of
and
are always , but they may have different domains.
,
, etc.


Generally,
.
Vertical line
Vertical line
0
0
0
is a function
is not a function
,
, and
FAQ (Frequently Asked Questions)
Sketch the graph of .
What is the domain/range of ?
Does
/ exist?
What is the minimum value of such that
which has domain
is one-one?
What is the minimum value of such that
exists, where has domain
?
Define
in a similar form.
Substitute rule, then sketch graph using GC.
Observe or sketch graph using GC.
Check conditions (one-one,
).
Note turning points or values where is
undefined.
Find the minimum value of
such that
.
Rule:
4. Let
.
5. Express in terms of , such that
.
6. Then
.
Domain
Define
in a similar form.
Rule:
Domain
Solve a general equation with functions.
Substitute rules, then solve or sketch graph
using GC.
Solve
or
instead.
 Solve
.
Both
and
have
rule
, so find the
 Solve
.
intersection of domains of and
.
Sketch the graphs of
,
, on a Show their geometrical relationship: graphs of
and
are reflections of each other about
single diagram.
.
Sketch the graph of
/
.
Graph of
is the section of
where
. (Similarly, graph of
is where
)
Find a function to model length, area, Use geometry (Pythagoras’ theorem, similar
volume etc.
triangles, etc.).
Example [NJC06H2/CT/Q9 (Modified)]
(a) A mapping is given by
,
.
(i)
(ii)
Explain why is not a function.
State the largest possible domain of in the form
, where
that the inverse function of exists. Hence define
in a similar form.
(b) The functions and are defined as follows:
, such
,
,
(i)
(ii)
Determine whether the composite function
exists.
Give the rule and domain of the composite function
and find its range.
Solution
(a)(i) Since the vertical line
, where
, does not cut the graph of at
one and only one point, is not a
function. █
(ii) Largest possible
Sketch the graph of on the GC. It may
appear to pass the vertical line test, but
and the graph indicate an
asymptote
undefined.
at
Use the GC to find
,
.
.
for all
.
(rejected)
,
(b)(i)
is
Use
to decide which of the 2
expressions for to accept and which to
reject. Here,
, so
,
and
or
so
exists if and only if is one-one.
Restrict based on the discontinuity at
, instead of looking for turning
points.
█
Let
,
█
Express
in a similar form, and write
down both its rule and domain.
has a minimum point at
and its domain
includes .
,
Since
,
the composite function
exist. █
, thus
does not
(ii)
exists if and only if
.
You do not need to show
exists, since
the question already assumes it.
,
█
is strictly increasing on
with
restricted to
█
.
In the restricted
,
decreases from to , reaches its
minimum point
, then
increases from
to
. Since
,
is the larger
endpoint of
.