Lecture 0
ECON 1202
UNSW Economics
c©UNSW Economics
Agenda
1 Function review;
2 Special functions;
3 Exponential and Logarithms;
4 Limits.
c©UNSW Economics
Definition: Function
A FUNCTION is a rule that assigns to each INPUT number exactly one
OUTPUT number.
c©UNSW Economics
Example (A linear function)
Consider what is meant by the simple linear function f (x) = 1+ 0.5x.
“ef-of-x is equal to 1 plus 0.5 times x.”
0 1 2 3 4 5 6
0
1
2
3
4
5
6
f (x)
x
*
*
*
*
*
*
c©UNSW Economics
Example (A linear function)
Consider what is meant by the simple linear function f (x) = 1+ 0.5x.
Each x value put into function f will
give exactly one f (x) value.
0 1 2 3 4 5 6
0
1
2
3
4
5
6
f (x)
x
*
*
*
*
*
*
c©UNSW Economics
Example (A linear function)
Consider what is meant by the simple linear function f (x) = 1+ 0.5x.
Each x value put into function f will
give exactly one f (x) value.
0 1 2 3 4 5 6
0
1
2
3
4
5
6
f (x)
x
*
*
*
*
*
*
c©UNSW Economics
Example (A linear function)
Consider what is meant by the simple linear function f (x) = 1+ 0.5x.
If we are dealing with CONTINUOUS
inputs (x can take any value between 0
and 5 say), then we normally draw a
(not necessarily straight) line to
represent the functional relationship
between x and f (x).
0 1 2 3 4 5 6
0
1
2
3
4
5
6
f (x)
x
c©UNSW Economics
1 The name of the function is irrelevant. For example,
f (x) =
x− 1
2
and g(z) = z + 1
are valid linear functions!
2 Often we talk in terms of DEPENDENT and INDEPENDENT
variables, or alternatively, in terms of the VALUE and ARGUMENT
respectively:
Example (Dependent, Independent)
Identify the dependent and independent terms, and the value and arguments
of the function H(a, b) = a2 + 2b + 3.
• H depends on a and b (the independent variables);
• H is the value, while a and b are the arguments.
c©UNSW Economics
1 A Function gives one output value for a given input value.
2 For this reason, they are also called a MAPPING, or a
TRANSFORMATION.
c©UNSW Economics
1 A Function gives one output value for a given input value.
2 For this reason, they are also called a MAPPING, or a
TRANSFORMATION.
Caution!
One of these is a function, one isn’t!
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = x2
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = ±√x
c©UNSW Economics
1 A Function gives one output value for a given input value.
2 For this reason, they are also called a MAPPING, or a
TRANSFORMATION.
Caution!
One of these is a function, one isn’t!
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = x2
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = ±√x
c©UNSW Economics
Definition: Domain and Range
The DOMAIN of a function is the set of all x values over which the
function ‘makes sense’ (works!). The RANGE of a function, is the set of
all possible f (x) values, given the domain.
Example
Find the domain of the function, y(x) = 2
x2+3x−4 .
Factorising gives,
y(x) =
2
(x− 1)(x + 4) ,
which implies, x 6= 1 or -4. Hence, the domain of y is the Real numbers except 1, -4.
c©UNSW Economics
Definition: The Constant Function
A CONSTANT FUNCTION is of the
form:
f (x) = c
where c is a constant.
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = 2
c©UNSW Economics
Definition: The Polynomial Function
A POLYNOMIAL FUNCTION is of
the form:
f (x) = cnx
n + cn−1xn−1+ · · ·+ c1x1+ c0
where cn . . . c0 are constants. -3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = x2
c©UNSW Economics
Definition: The Rational Function
A RATIONAL FUNCTION is of the
form:
f (x) =
p1(x)
p2(x)
where p1 and p2 are polynomial
functions.
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = x
2−6
x+6
c©UNSW Economics
Definition: The Absolute Function
An ABSOLUTE VALUE FUNCTION is
of the form:
f (x) = |g(x)|
where g(x) is some function and
| · | indicates ‘positive value’. -3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = | − √x|
c©UNSW Economics
Suppose we have two functions, f (x) = 3x + 2 and p(x) =
√
x− 1,
then we can create a new function using standard operations:
SUM f (x) + p(x)
DIFFERENCE f (x)− p(x)
PRODUCT f (x) · p(x)
QUOTIENT
f (x)
p(x)
Caution!
The domain of a combined function is the set of all x such that f (x) and
p(x) are well defined, and the denominator is not zero for the quotient.
c©UNSW Economics
Example (Combining functions)
Suppose f (x) = 2x2 − 3x− 2 and g(x) = x− 2, and let h(x) = f (x)
g(x)
, then show
that h(x)− g(x) = x + 3.
We begin by finding h(x),
h(x) =
f (x)
g(x)
=
2x2 − 3x− 2
x− 2 =
(2x + 1)(x− 2)
x− 2 = 2x + 1 ,
now,
h(x)− g(x) = 2x + 1− (x− 2) = x + 3 .
c©UNSW Economics
Now suppose we don’t want a combination, but we want to construct
a process of more than one function,
x f (x) y g(y) z
that is,
x h(x) z
Definition: Composite Function
If f and g are functions, the COMPOSITE FUNCTION of g and f is the
function g ◦ f ,
(g ◦ f )(x) = g(f (x)) ,
and the domain of g ◦ f is the values of x in the domain of f such that
f (x) is in the domain of g.
c©UNSW Economics
Example (Composite functions)
Let p(x) = x2 − 2, and h(x) = √5x + 1 (for x ≥ 0). Find (p ◦ h)(2).
We have,
(p ◦ h) = p(h(x))
solving by substitution,
p(h) = h2 − 2
∴ p(h(x)) = (
√
5x + 1)2 − 2
∴ p(h(2)) = (5)(2)− 1 = 9
c©UNSW Economics
Now suppose that instead of,
x f (x) y
we want to go back the other way, that is,
x ? y
or in other words, if
f (x) = y ,
then what we are after is the function,
f−1(y) = x .
where f−1 is the INVERSE FUNCTION of f .
c©UNSW Economics
Example
Suppose f (x) = x2 + 1, find f
−1.
First, let
y = f (x) =
x
2
+ 1 ,
now, solve for x in terms of y,
y− 1 = x
2
2(y− 1) = x
f−1(y) = 2(y− 1)
c©UNSW Economics
Caution!
A function may not have an inverse function
Example
Suppose f (x) = x
2+1
5 , find f
−1(x).
First, let
y = f (x) =
x2 + 1
5
,
now, solve for x in terms of y,
∴ 5y− 1 = x2
∴ x = ±√5y− 1
f−1(y) = ±√5y− 1
c©UNSW Economics
Let’s try that out, suppose x = 2:
f (x) = f (2) =
(2)2 + 1
5
= 1
... and the other way around,
f−1(x) = f−1(1) = ±
√
(5)(1)− 1 = ±2
???!!! we received two answers back: +2, or −2
Caution!
A function has an inverse if and only if it is a ONE-TO-ONE function.
Definition: One-to-one Function
A function is one-to-one if for all a and b, if a 6= b, then f (a) 6= f (b).
c©UNSW Economics
Caution!
Do not confuse the inverse and reciprocal of a function.
For instance, to represent the reciprocal of the function f (x) we would
normally write,
[f (x)]−1 =
1
f (x)
,
rather than when we just want the inverse we would write,
f−1(x) .
c©UNSW Economics
Example
Suppose f (x) = x2 + 1, find f
−1(x) and [f (x)]−1.
To find the inverse f−1(x), we solve f (x) = y for x in terms of y
y =
x
2
+ 1 =⇒ x = 2(y− 1) =⇒ f−1(y) = 2(y− 1) .
To find the reciprocal [f (x)]−1, we simply divide 1 by f (x)
[f (x)]−1 = 1
f (x)
=
1
x
2 + 1
=
2
x + 2
.
c©UNSW Economics
Definition: Exponential
f (x) = ax
(A selection of) Important rules:
aman = am+n
am
an
= am−n
(am)n = amn
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = 2x
c©UNSW Economics
Definition: Logarithmic
Where b is the BASE,
f (x) = logb x
(A selection of) Important rules:
logb(mn) = logb m + logb n
logb(
m
n
) = logb m− logb n
logb(m
r) = r logb m
logb 1 = 0
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x
f (x) = log10 x
Note: logbx is like saying, ‘what power must I raise b to, to obtain x?’
c©UNSW Economics
The connection between LOGARITHMS and EXPONENTIALS...
Definition: A very nice rule
logb x = y corresponds to b
y = x
log
b
x = y
c©UNSW Economics