Introduction
The concept of a function is fundamental to many of the applications that we
will encounter in economics. As we have already seen in Chapters 2 and 3, it is a
convenient way of expressing a relationship between two variables in terms of a
prescribed mathematical rule. More formally, we have the following definition:
Definition 4.1
A function f is a rule that assigns to each value of a variable x, called the
independent variable of the function, one and only one value f(x), referred
to as the value of the function at x. The variable y = f(x) varies with x
and is known as the dependent variable.
We sometimes write f(x) to denote the function f if we wish to indicate
that the variable is x. The function rule defines the dependent variable in terms
of the independent variable. A function of a single variable enables the value
of the dependent variable to be determined when the independent variable is
specified. A function may therefore be interpreted as a process f that takes
an input number x and converts it into only one output number f(x). For
example, the function defined by the rule f(x) = 6x + 2 is the rule that takes
an input number x, multiplies it by 6, and then adds 2 to the product to obtain
the output number. Given a value of x, the corresponding value of f(x) can be
69
70 Elements of Mathematics for Economics and Finance
determined using this rule. For example, if x = 3
f(x) = 6× 3 + 2 = 18 + 2 = 20.
We write f(3) = 20 and say ‘the value of f at x = 3 is 20’ or ‘f of 3 equals 20’.
Example 4.2
Evaluate f(x) = 2x −5 when x = −1, x = 2 and x = 4.
Solution. When x = −1,
f(x) = 2 × (−1) −5 = −2 −5 = −7,
so that f(−1) = −7. When x = 2,
f(x) = 2 × 2 − 5 = 4 −5 = −1,
so that f(2) = −1. When x = 4,
f(x) = 2 × 4 − 5 = 8 − 5 = 3,
so that f(4) = −3.
Functions are generally represented by algebraic formulae that are usually
expressed in the form
y = f(x),
where f defines the precise nature of the functional relationship. We say ‘y
equals f of x’ or ‘y is a function of x’. In mathematics, we usually denote
functions by letters such as f, g, and h. Examples of functions are :

1. the linear function y = f(x) = ax + b;
2. the quadratic function y = f(x) = ax2 + bx + c;
3. the power function y = f(x) = axn;
   where a, b, c and n are constants.
   Example 4.3
   Given f(x) = x2 + 4x − 5, find f(2) and f(−3).
4. Functions of a Single Variable 71
   Solution.
   f(2) = 22 + 4(2) − 5 = 4+8 − 5 = 7
   f(−3) = (−3)2 + 4(−3) − 5 = 9 − 12 −5 = −8
   There are occasions when the input number or value of the dependent variable
   is not admissible in the sense that the function fails to process it. For
   example, take the reciprocal function f(x) = 1/x and consider the input value
5. If we try to evaluate f(0) on a calculator, an error message will be given
   because we cannot divide by zero. Some calculators will even deliver a reprimand
   and inform you that you cannot divide by zero! All the numbers that a
   function can process are known collectively as the domain of the function.
   Sometimes we may wish to restrict the domain to a smaller set of numbers
   than are admissible. In many applications in economics, we are only interested
   in domains that contain nonzero numbers. For example, the profit function
   is only of interest for non-negative values of output even though it may well
   be defined for negative values as well. The smaller set of numbers is called a
   restricted domain. For example, the function defined by
   f(x) = 2x + 1, −2 ≤ x ≤ 4, (4.1)
   has a domain restricted to all the real numbers lying between −2 and 4 even
   though this function is defined over all the real numbers. The range of a
   function is the collection of all those values of f(x) that correspond to each
   and every number in the domain of the function. For example, the function
   f(x) = x2 has a domain that consists of all the real numbers and a range that
   contains all the non-negative real numbers. The function f(x) defined by (4.1)
   has domain −2 ≤ x ≤ 4 and range −3 ≤ f(x) ≤ 9.
   Note that a function can take the same value for two different values of its
   argument. For example, the function f(x) = x2 takes the value 4 when x = −2
   and x = 2. Such functions are said to be many-to-one. Functions that are
   such that each element x of the domain is assigned to a different value f(x)
   are said to be one-to-one, i.e., the function f is one-to-one if
   f(x1) = f(x2) implies x1 = x2.
   Every linear function
   f(x) = ax + b, a = 0,
   is one-to-one. Relationships which are one-to-many can occur, but from our
   definition they are not functions. For example, y2 = 1 − x2 is an example of
   a one-to-many relationship. When x = 0, y2 = 1, and so y = −1 and y = 1.
   Therefore, there are two values of y that correspond to x = 0.
   72 Elements of Mathematics for Economics and Finance
   4.2 Limits
   Sometimes it is of interest to know how a function behaves as the value of its
   argument tends to a fixed value. For example, in economics one may wish to
   know how the average cost of producing a certain good decreases as the number
   of goods produced increases. For example, suppose that the total cost to an
   electronics company of producing Q flat screen televisions is
   TC = 800Q + 1,000,000.
   What is the average cost AC of producing Q televisions when Q is very large?
   We can answer questions such as this using the concept of a limit.
   The limiting behaviour of a function when the values in its domain are
   larger than any finite number may be formalised by expressing the limit of a
   function f(x) as x moves increasingly far to the right on the real line as
   lim
   x→∞f(x).
   So x→∞ means x increases without bound, and we say x tends to ∞. Similarly,
   the limit of f(x) as x moves increasingly far to the left on the real line is
   expressed as
   lim
   x→−∞f(x).
   So x → −∞ means x decreases without bound, and we say x tends to −∞.
   In the next section, the concept of the limit of a function will be explored and
   explained for the reciprocal function f(x) = 1/x.
   4.3 Polynomial Functions
   The properties of linear and quadratic functions were described in Chapters 2
   and 3, respectively. In this section, we look at other polynomial functions. First
   of all, consider the power functions defined by
   f(x) = xn,
   where n is a positive integer. These are sometimes known as monomials since
   they comprise only one term.
   If n is even, the graph of f(x) = xn is similar to that of f(x) = x2 in terms
   of its shape and its symmetry about the y-axis (see Fig. 4.1). The important
   difference is that, for n > 2, f(x) increases more rapidly as x increases away
   from x = ±1 in the positive and negative x-directions. Note that all the graphs
6. Functions of a Single Variable 73
   pass through the three points (0, 0) (where they attain their minimum values),
   (1, 1) and (−1, 1).
   If n is odd, the graphs of f(x) = xn are similar, for positive values of x,
   to those for which n is even. However, for negative values of x they are quite
   different (see Fig. 4.2). The portion of the graph for negative values of x may
   be formed as the result of two reflections of the positive portion of the graph,
   first with respect to the y-axis and then with respect to the x-axis, i.e., if the
   point (x, y) lies on the graph then so also does the point (−x,−y). All the
   graphs pass through the points (0, 0), (1, 1), and (−1,−1).
   If n is odd, the function f(x) = xn is an increasing function of x since
   f(x1) ≤ f(x2) for x1 < x2. If n is even, the function f(x) = xn is an increasing function of x for x ≥ 0. However, for x ≤ 0, the function is decreasing since f(x1) ≥ f(x2) for x1 < x2 ≤ 0. The general cubic function has the form f(x) = ax3 + bx2 + cx + d, -1 0 1 5 10 15 n=2 n=4 n=6 y x Figure 4.1 The graphs of the even monomials f(x) = xn for n = 2, 4, 6. 74 Elements of Mathematics for Economics and Finance -2 -1 0 1 2 -10 -5 5 10 n=1 n=3 n=5 y x Figure 4.2 The graphs of the odd monomials f(x) = xn for n = 1, 3, 5. with a = 0. The simplest cubic function is f(x) = x3. Its graph is the green curve in Fig. 4.2. More generally, the graph of a cubic function has one of the two forms shown in Figs. 4.3 and 4.4 depending on the sign of a. If a > 0, f(x)
   tends to ∞ as x tends to ∞ and tends to −∞ as x tends to −∞. The cubic
   function f(x) = x3 +x2 −2x has a = 1 > 0, and its graph is shown in Fig. 4.3.
   If a < 0, f(x) tends to −∞ as x tends to ∞ and tends to ∞ as x tends to −∞.
   The cubic function f(x) = −x3 +5×2 −2x−15 has a = −1 < 0, and its graph
   is shown in Fig. 4.4.
   The graph of a cubic function crosses the x-axis at one, two, or three points.
   Therefore, the equation f(x) = 0 has one, two, or three real roots. For example,
   the graph of the cubic function f(x) = x3 + x2 − 2x (see Fig. 4.3) crosses the
   x-axis when x = −2, x = 0 and x = 1, and the graph of the cubic function
   f(x) = −x3 +5×2 −2x−15 (see Fig. 4.4) crosses the x-axis at the single point
   x = −3/2. The graph of the function f(x) = x3−x2 crosses the x-axis at x = 1
   and x = 0. At x = 0 the function f(x) = x3 − x2 has two coincident roots.
7. Functions of a Single Variable 75
   -3 -2 -1 0 1 2
   -15
   -10
   -5
   5
   10
   y 15
   x
   Figure 4.3 The graph of the function f(x) = x3 + x2 − 2x.
   4.4 Reciprocal Functions
   Consider the reciprocal function defined by
   f(x) =
   1
   x
   ,
   for x > 0. All the applications considered in this book are for x > 0. However,
   for completeness we also sketch the function for x < 0 in Fig. 4.5. Here we see that the part of the graph for x < 0 is obtained by reflecting the graph for x > 0 in the line y = −x. As we have already noted, this function is not defined
   for x = 0.
   When x is large and positive, f(x) is small and positive, and as x takes
   increasingly larger values, f(x) takes values that approach but never reach 0.
   For example, f(10) = 0.1, f(100) = 0.01, f(1,000) = 0.001, etc. As x → ∞,
   the graph of f(x) gets arbitrarily close to the x-axis and therefore
   lim
   x→∞
   1
   x
   = 0.
   When x is large and negative, f(x) is small and negative. As x → −∞, the
   graph of f(x) gets arbitrarily close to the x-axis approaching it from below and
   76 Elements of Mathematics for Economics and Finance
   -2 2 4
   -20
   -15
   -10
   -5
   0
   5
   10
   15
   20
   y 25
   x
   Figure 4.4 The graph of the function f(x) = −x3 + 5×2 − 2x − 15.
   therefore
   lim
   x→−∞
   1
   x
   = 0.
   The idea of a limit can also be used to describe the unbounded behaviour of
   functions. For example, consider the limit of the reciprocal function f(x) = 1/x
   for x = 0 as x tends to 0 from the right (see Fig. 4.5), i.e., x takes only positive
   values. As x takes increasingly smaller values, f(x) takes increasingly larger
   values. For example, f(1) = 1, f(0.1) = 10, f(0.01) = 100, f(0.001) = 1,000,
   etc. The values of f are positive and become arbitrarily large in this limit, i.e.,
   given any positive number y we can always find a value of x for which f(x) > y.
   We express this mathematically as
   lim
   x→0+
   1
   x
   = ∞.
   The superscript ‘+’ on 0 indicates that we are taking the limit as x approaches
   0 from the right through positive values. Similarly, the values of f(x) as x tends
   to 0 from the left are negative and become arbitrarily large in this limit, i.e.,
   given any negative number z we can always find a value of x for which f(x) < z.
   We express this mathematically as
   lim
   x→0−
   1
   x
   = −∞.
8. Functions of a Single Variable 77
   -4 -2 2 4
   -4
   -2
   0
   2
   4
   y
   x
   Figure 4.5 The graph of the function f(x) = 1/x.
   The superscript ‘−’ on 0 indicates that we are taking the limit as x approaches
   0 from the left through negative values.
   Example 4.4
   The fixed costs of producing a good are 10 and the variable costs are 4 per unit.
   Find expressions for total cost TC and average cost AC. Sketch the graph of
   AC as a function of Q.
   Solution. The total cost function is
   TC = FC + V C × Q
   = 10+4Q.
   The average cost function, AC, is given by
   AC = TC
   Q
   .
   78 Elements of Mathematics for Economics and Finance
   Table 4.1 Tables of values of AC in Example 4.4.
   Q 0.01 0.1 1 10 100
   AC 1004 104 14 5 4.1
   Therefore, using the above expression for TC we have
   AC =
   10 + 4Q

Q

10
Q
+
4Q

Q

10
Q

• 4.
  This function is tabulated in Table 4.1 and sketched in Fig. 4.6. The dashed
  line in this figure corresponds to V C = 4. As Q tends to ∞, AC tends to 4,
  i.e.,
  lim
  Q→∞
  AC = 4.
  In this example, it is no coincidence that AC approaches the value of V C,
  i.e., 4, as Q becomes large. In fact, this result holds whenever V C is constant.
  To see this, let us examine the expression for AC:
  AC = TC
  Q
  = FC + V C × Q
  Q
  = FC
  Q
• V C.
  As Q becomes large, FC/Q approaches 0. Therefore, AC tends to V C as Q
  tends to ∞, i.e.,
  lim
  Q→∞
  AC = V C.
  Example 4.5
  The fixed costs of producing a good are 8, and the variable costs are 3 + 5Q
  per unit. Find expressions for total cost TC and average cost AC. Evaluate
  TC and AC when Q = 10. Sketch the graph of AC as a function of Q.

1. Functions of a Single Variable 79
   5 10 15
   0
   10
   20
   30
   40 AC
   VC
   AC, VC
   Q
   Figure 4.6 The graphs of the average cost function AC = 10
   Q +4 and the
   variable cost per unit V C = 4.
   Solution. The total cost function is
   TC = FC + V C × Q
   = 8+(3+5Q)Q
   = 8+3Q + 5Q2.
   The average cost function, AC, is given by
   AC = TC
   Q
   .
   Therefore, using the above expression for TC we have
   AC =
   8 + 3Q + 5Q2

Q

8
Q
+
3Q
Q
+
5Q2

Q

8
Q

• 3 + 5Q.
  When Q = 10,
  TC = 8+3 × 10 + 5 × 102 = 8 + 30 + 500 = 538,
  80 Elements of Mathematics for Economics and Finance
  Table 4.2 Tables of values of AC in Example 4.5.
  Q 0.01 0.1 1 10 100
  AC 803.05 83.5 16 53.8 503.08
  and
  AC =
  8
  10
• 3 + 5 × 10 = 0.8 + 3 + 50 = 53.8.
  This function is tabulated in Table 4.2 and sketched in Fig. 4.7. The dashed
  line in this figure is the straight line AC = 3+5Q. As Q tends to ∞, AC
  tends to V C. This is because the term 8/Q in the equation for AC becomes
  negligibly small for large values of Q. Since V C tends to ∞ as Q tends to ∞,
  we have
  lim
  Q→∞
  AC = ∞.
  5 10 15
  0
  20
  40
  60
  80
  AC
  VC
  AC, VC
  Q
  Figure 4.7 The graphs of the average cost function AC = 8
  Q + 3 + 5Q
  and the variable cost per unit V C = 3+5Q.

1. Functions of a Single Variable 81
   Example 4.6
   Suppose that the total cost to an electronics company of producing Q flat screen
   televisions is
   TC = 800Q + 1,000,000.
   Obtain an expression for the average cost function. What is the average cost
   of production when Q is very large?
   Solution. The average cost function is given by
   AC = TC

Q

800Q + 1,000,000
Q
= 800 +
1,000,000
Q
.
The second term in this expression for the average cost function tends to 0 as
Q tends to ∞. Therefore, in the limit of arbitrarily large Q we have
lim
Q→∞
AC = 800.
4.5 Inverse Functions
Given a function y = f(x), consider the reverse process in which y becomes
the input and x the output. This reverse process, under certain conditions,
defines what is known as the inverse function of f. If we denote the inverse
function by, say g, then we can write x = g(y) (see Fig. 4.8). Thus, y is now the
independent variable and x the dependent variable. For example, consider the
determination of the inverse of the function y = f(x) = 6x+2. This is achieved
by reversing the input and output processes of the function. The inverse of the
function that multiplies the input by 6 and then adds 2 to the result is the
process that subtracts 2 from the input and then divides the result by 6. This
process defines the inverse of the function, i.e.,
x = g(y) = y − 2
6 .
The inverse of a one-to-one function satisfies the definition of a function
and therefore is itself a function. Therefore, a necessary condition for a given
function to have an inverse is that it is one-to-one. Thus every linear function
f(x) = ax + b, a = 0, has an inverse since it is one-to-one.
Nonlinear functions may not possess an inverse function. For example, the
function
y = f(x) = x2,
82 Elements of Mathematics for Economics and Finance
y
g(y)
(g(y),y)
0
Figure 4.8 Graph of y = f(x) where g is the inverse function of f.
is a many-to-one function, i.e., there are two values of x that correspond to
each value of y (see Fig. 4.9 where x = ±2 both correspond to y = 4). If
we tried to find the inverse of this many-to-one function, we would obtain a
one-to-many relationship, which contravenes the definition of a function. Thus,
only a one-to-one function can possess an inverse. However, if the domain of f
is restricted to positive values of x, say, then f does possess an inverse defined
by
x = g(y) =
√
y.
This situation is shown in Fig. 4.10.
Example 4.7
Find the inverses of the functions

1. f(x) = 2x − 3,
2. f(x) = (x − 2)2, 2 ≤ x.
3. Functions of a Single Variable 83
   -3 -2 -1 0 1 2 3
   2
   4
   6
   8 y=x2
   y
   x
   Figure 4.9 Graph showing the many-to-one function y = x2.
   Solution.
4. Let y = 2x − 3. We rearrange this equation so that x appears by itself on
   the left-hand side. Adding 3 to both sides, we have
   y + 3 = 2x.
   Finally, dividing both sides by 2 yields the inverse function
   x = g(y) = y + 3
   2 .
5. Let y = (x − 2)2. For x ≥ 2 this function is one-to-one and therefore
   possesses an inverse. Taking the square root of both sides of this equation
   gives
   x −2 =
   √
   y.
   Finally, adding 2 to both sides yields the inverse function
   x = g(y) =
   √
   y + 2.
   84 Elements of Mathematics for Economics and Finance
   y=x2
   y
   √ y
   _
   0 x
   Figure 4.10 Graph showing the inverse function of f(x) = x2 when the
   domain of f is restricted to positive values of x.
   The motivation for introducing inverse functions in this book is that in
   economics, some functions are plotted with the dependent variable y on the
   horizontal axis and the independent variable x on the vertical axis. The demand
   function is one such example. The demand function expresses the dependence
   of the quantity demanded, Q, of a good on the market price, P. We may write
   this function as
   Q = f(P).
   Given a particular rule for f, it is relatively simple to determine the value of Q
   for a given value of P or to sketch the graph of the function. A mathematician
   would plot this function with the independent variable (P) on the horizontal
   axis and the dependent variable (Q) on the vertical axis. However, economists
   prefer to plot them the other way round with Q on the horizontal axis and P
   on the vertical axis. To facilitate this, the demand equation is rearranged so
   that P is expressed in terms of Q, i.e.,
   P = g(Q),
   for some function g. The functions f and g are said to be inverse functions.
6. Functions of a Single Variable 85
   Example 4.8
   For the demand function Q = f(P), where
   f(P) = −P
   3

• 18
  determine the value of Q when P = 30. Express P in terms of Q and hence
  find the value of P when Q = 9.
  Solution.
  Q = f(P) = −P
  3
• 18.
  When P = 30,
  Q = −30
  3
• 18 = −10 + 18 = 8.
  To express P in terms of Q, we rearrange the terms to isolate P on the left-hand
  side of the equation. Multiplying both sides by 3 gives
  3Q = −P + 54.
  A simple rearrangement of this equation yields the following expression for P
  in terms of Q
  P = −3Q + 54.
  When Q = 9,
  P = −3(9) + 54 = −27 + 54 = 27.
  In determining P as a function of Q, we have found the inverse function of
  f. We may write
  P = g(Q), where g(Q) = −3Q + 54.
  EXERCISES
  4.1. Sketch the graph of the cubic function f(x) = 6+12x + 3×2 − 2×3
  for −2 ≤ x ≤ 3.
  4.2. Sketch the graph of the cubic function f(x) = 8×3 +30×2 +13x−15
  for −4 ≤ x ≤ 2.
  4.3. The fixed costs of producing a good are 12 and the variable costs
  are 7 per unit. Find expressions for TC and AC. Evaluate TC and
  AC when Q = 4 and Q = 12. Sketch the graph of AC as a function
  of Q.
  86 Elements of Mathematics for Economics and Finance
  4.4. The fixed costs of producing a good are 9 and the variable costs are
  4+3Q per unit. Find expressions for TC and AC. Evaluate TC and
  AC when Q = 5 and Q = 10. Sketch the graph of AC as a function
  of Q.
  4.5. Suppose that the total cost to a furniture company of producing Q
  desks is
  TC = 50Q + 40,000.
  Obtain an expression for the average cost function, AC. What value
  does AC approach when Q is very large?
  4.6. Find the inverses of the following functions:
  a) f(x) = −3x + 2,
  b) f(x) = 5x + 3,
  c) f(x) = (x − 3)2 + 2, 3 ≤ x.
  4.7. For the demand function
  Q = −P
  4
• 25
  determine the value of Q when P = 36. Express P in terms of Q
  and hence find the value of P when Q = 5.