Chapter1: Functions
Functions topics: 1.1 Simple Algebraic and Trigonometric Functions; 1.2 Graphs of Simple
Functions [4 HOURS]
CONTENTS
1.0 Introduction-Function ...................................................................................................... 1
1.1 Definition of a Function ................................................................................................... 1
SPECIAL FUNTIONS........................................................................................................... 3
1.2 Absolute-Value Function ................................................................................................. 3
1.3 Linear Function ................................................................................................................ 3
1.4 Composite Function ......................................................................................................... 3
1.5 Rational Function ............................................................................................................. 3
1.6 Quadratic Function........................................................................................................... 4
1.7 Polynomials...................................................................................................................... 4
1.8 Exponential Functions ..................................................................................................... 4
1.9 The Natural Exponential Function ................................................................................... 5
1.10 The Trigonometric Functions ........................................................................................ 6
2.1 Application of Functions.................................................................................................. 7
3.1 Graphs of simple functions .............................................................................................. 7
REFERENCES .................................................................................................................... 10
EXERCISES ........................................................................................................................ 10
TEST AND EXAM QUESTIONS ...................................................................................... 11
1.0 Introduction-Function
A function is a special type of input-output relation that expresses how one quantity (output)
depends on another quantity (input).
To illustrate, suppose P100 earns simple interest at an annual rate of 6%. Then it can be
shown that interest are related by the formula
I = 100 (0.06) t
where I is interest in Pula and t is the time in years. For example,
If t=1/2,
then I=100(0.06)(1/2) =3
We can think formula (1) as defining a rule. This rule is an example of function.
1.1 Definition of a Function
.
Domain
.
Range
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(1)
(2)
A function is a rule that produces a relation between two variables by an equation. We get
exactly one output (value for the dependent variable) for each input (value for the
independent variable).
A function links each member of the domain to only one member of the range. y = f ( x ) is a
function where x is an independent variable and y is a dependent variable.
Remark 1.1: If we get more than one output for a given input, then the equation does not
define a function.
Mathematical Definition of Function: A function f with domain A is the set of all ordered
pairs ( x, f ( x)) where x belongs to A.
Input → f(x) →output
Domain and Range
Definition: The set of input values is called the domain and the set of output values is called
the range.
Example 1.1.1
Consider a rectangle with length x cm and width ( x − 1) cm. If y denotes the area of the
rectangle, then the equation is y = x ( x − 1) , x > 1 . For each input x , we obtain an output y
(= area). Here x is an independent variable and y is dependent. Here the domain is x > 1 ,
i.e. (1, ∞ ) and the range is y > 0 , i.e. (0, ∞ ).
Example 1.1.2
Suppose y = 5 x − 1 .
If x = 2 ,
then y = 5 ( 2 ) − 1 = 10 − 1 = 9 .
If x = −2 ,
y = 5 ( −2 ) − 1 = −10 − 1 = −11 . Here the domain is − ∞ < x < ∞ , i.e, (−∞, ∞)
then
and the range
is − ∞ < y < ∞ , i.e, (−∞, ∞) .
Example 1.1.3 (Not a function)
y 2 − x 2 = 9 is not a function.
y 2 = 9 + x 2 ⇒ y = ± 9 + x 2 . We are getting two values of y for every value of x . Hence
y 2 − x 2 = 9 is not a function (see Remark 1.1)
Notation:
We use f , g , h, etc. to denote functions.
For example: y = f ( x ), y = g ( x ), y = h( x ), etc.
In Example 1.1.1, f ( x ) = x(x − 1) , while in Example 1.1.2, f ( x ) = 5 x − 1 .
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STA102: Mathematics for Business and Social Sciences II
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SPECIAL FUNTIONS
1.2 Absolute-Value Function
The function f(x) = |x| is called the absolute value function.
The absolute value or magnitude of a real number x is denoted by |x| and is defined by:
x , x ≥ 0 ,
x =
 − x , x < 0.
Thus the domain of f is all real numbers.
1.3 Linear Function
A linear function is a function of the form y = f ( x ) = mx + c , where m ≠ 0 and c are
constants. It represents a straight line. c is the y - intercept. It is obtained by putting x = 0
in y = mx + c . It occurs at the point ( 0, c ) . m represents the slope of the line. It is the rate of
change in y that accompanies a unit change in x. That is, if x changes by one unit, then y
changes by m units.
Example 1.3.1
Consider the line y = 4 x − 3 . If x = 2 , then y = 4 ( 2 ) − 3 = 8 − 3 = 5 . Now suppose x changes
by one unit from 2 to 3. In this case y = 4 ( 3) − 3 = 12 − 3 = 9 . Notice that as x changes by one
unit from 2 to 3, y changes by 4 units from 5 to 9.
1.4 Composite Function
If f ( x ) and g ( x ) are two functions of x , then f ( g ( x )) is called the composite function. We
obtain f ( g ( x )) by replacing x in f ( x ) by g ( x ) .
Example 1.4.1
If f ( x ) = 3x − 6 and g ( x ) = x , obtain the composite functions f ( g ( x )) and g ( f ( x )) .
Solution 1.4.1
f ( g ( x ) ) = 3 x − 6 = 3 x − 6 and g ( f ( x )) = 3 x − 6
( )
1.5 Rational Function
g ( x)
, h ( x ) ≠ 0 where g ( x ) and h( x ) are polynomial
h ( x)
functions of x , are called rational functions.
Functions of the form y = f ( x ) =
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For example, f ( x ) =
x 2 + 2x − 2
is a rational function.
x−5
1.6 Quadratic Function
A function of the form y = f ( x ) = ax 2 + bx + c is called a quadratic function, where a ≠ 0, b
and c are constants. The graph of the quadratic function is a parabola.
The parabola opens up (convex from below) if a > 0 and opens down (concave from below)
if a < 0.
The roots of the function are determined by solving the equation y = ax 2 + bx + c = 0 .
−b
If a > 0 , the minimum occurs at x =
2a
−b
If a < 0 , then the maximum occurs at the point where x =
.
2a
−b
A parabola is symmetric about the vertical line x =
.
2a
Note: You should be able to sketch the graph of a quadratic function.
1.7 Polynomials
A function of the form f ( x ) = an x n + an −1 x n −1 + + a1 x + a0 , an ≠ 0 is called a polynomial of
degree n .
y = 3 x 3 + 5 x + 8 is a polynomial of degree 3.
f ( x ) = 5 x 2 + 3 x + 2 is a polynomial of degree 2 (quadratic function).
f ( x ) = 3x + 4 is a polynomial of degree 1 (linear function).
Example 1.7.1
Graph the function y = f ( x ) = 2 x 3 − 3 x 2 − 12 x − 1
1.8 Exponential Functions
An exponential function has the form f ( x ) = b x , b > 0 and b ≠ 1 , where b is constant and it
is called the base. The variable x can be any real number, i.e. − ∞ < x < ∞ . The domain of
f ( x ) is − ∞ < x < ∞ and the range is 0 ≤ y < ∞ where y = f ( x ) .
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Example 1.8.1
x
1 1
x
The functions f ( x ) = 2 and g ( x ) = 2 = x =   = (0.5) are both exponential functions.
2
2
 
The base of f ( x ) is 2 and the base of g ( x ) is 0.5.
x
−x
Example 1.8.2 (Not an exponential function)
1
The functions h( x ) = x 2 and g ( x ) = x − 2 = 2 are not exponential functions. They are only
x
functions because the base is x which is variable.
Example 1.8.3 (Not an exponential function)
x
The function f ( x ) = (− 3) is not an exponential function because the base is -3 which is
negative.
Example 1.8.4 (Not an exponential function)
The function f ( x ) = 1x is not an exponential function because the base is 1.
Remark:
(1) The exponential function f ( x ) = b x increases as x increases if b > 1 . For example
f ( x ) = 2 x increases as x increases. That is f (1) = 21 = 2 , f ( 2 ) = 22 = 4 , f ( 3) = 23 = 8 and
so on.
(2) The exponential function f ( x ) = b x decreases as x increases if 0 < b < 1 .
For example
2
f ( x ) = (0.5)
x
x increases. That is
decreases as
1
f (1) = ( 0.5 ) = 0.5 ,
3
f ( 2 ) = ( 0.5 ) = 0.25 , f ( 3) = ( 0.5 ) = 0.125 and so on.
1.9 The Natural Exponential Function
The function f ( x ) = e x is called the natural exponential function where e = 2.71828… which
is an irrational number. The domain is (− ∞, ∞ ) and the range is (0, ∞ ) .
Example 1.9.1
Following are the examples of natural exponential functions:
2
g (x ) = e − x
and
(c)
h( x ) = e − x
(a) f ( x ) = e x , (b)
Example 1.9.2
The population of a certain city during the years 1950 to 2000 can be modeled by the
function p(t ) = 10000e 0.05t , 0 ≤ t ≤ 50 , where t = number of years elapsed after 1950 and
p(t ) = population of the city at the t th year.
Obtain p(10) . (Exercise for student)
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STA102: Mathematics for Business and Social Sciences II
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Example 1.9.3
Rewrite the equations
(a) y = 8.2(2.5) x
(b) y = 12(0.4) x
in the form y = aebx , where a and b are constants.
Solution 1.9.3
(a) y = 8.2(2.5) x
(b)
= 8.2 e x ln 2.5
= 8.2 e0.916 x
y = 12(0.4) x
= 12 e x ln 0.4
= 12 e−0.916 x
1.10 The Trigonometric Functions
In trigonometry, the angles are measured either in degrees or radians. A complete revolution
encompasses 360 degrees or 2 π radians. Thus π radians = 180i .
We consider three basic trigonometric functions defined by
(a) f ( x) = sin x
(b) f ( x) = cos x
(c) f ( x) = tan x
Here the angle x is measured in radians. The trigonometric functions are related to the sides
of a right angled triangle as follows.
B
x
A
sin x =
C
AB
AC
AB
, cos x =
and tan x =
BC
BC
AC
Table of Signs of trigonometric functions:
Function/Quadrant
sinx
cosx
tanx
I
+
+
+
II
+
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STA102: Mathematics for Business and Social Sciences II
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III
+
IV
+
-
2.1 Application of Functions
2.1.1 Life Sciences
Example 2.1.1 (Physiology)
A good approximation of the normal weight of a person 60 inches tall but not taller than 80
inches is given by w( x ) = 5.5 x − 220, where x is height in inches and w( x ) is weight in
pounds. Find the domain and range.
Solution 2.1.1
The equation of the function is w( x ) = 5.5 x − 220, 60 ≤ x ≤ 80 .
Domain: 60 ≤ x ≤ 80
Range: 110 ≤ w( x ) ≤ 220
2.2.2 Business and Economics
Example 2.2.2 (Price demand)
The retail price p( x ) and the weekly demand x for a particular model CD players are related
by p ( x ) = 115 − 4 x , 9 ≤ x ≤ 289 . Obtain the domain and the range.
Solution 2.2.2
Domain: 9 ≤ x ≤ 285 and Range: 47 ≤ p( x ) ≤ 103
3.1 Graphs of simple functions
Definition: The graph of a function f is the set of all points ( x, y ) in the xy − plane such
that x is the domain and y = f ( x) is the range.
Example 3.1.1
Draw the graph of the following function: y = 2 x + 1 .
Solution 3.1.1
Point 1: (0, 1)
Point 2: (-0.5, 0)
X
-1
Y=2X+1
-1.0
-0.5
0.0
0
1.0
0.5
2.0
1
3.0
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STA102: Mathematics for Business and Social Sciences II
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Example 3.1.2
Draw graph of the function y = x 2 , − ∞ < x < ∞ .
Solution 3.1.2
x
y=x
2
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
36
25
16
9
4
1
0
1
4
9
16
25
Example 3.1.3
Draw the graph of the following absolute value function y = x .
Solution 3.1.3
 x , x ≥ 0,
y= x =
− x , x < 0.
x
y=|x|
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
6
5
4
3
2
1
0
1
2
3
4
5
6
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STA102: Mathematics for Business and Social Sciences II
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Example 3.1.4
Draw the graph of the function y = f ( x ) = x − 2 .
Solution 3.1.4
x − 2, x ≥ 0,
y = x−2 = 
 − x − 2 , x < 0.
x
y=|x-2|
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
Vertical Line Test:
A curve in the xy − plane is the graph of a function y = f ( x) if and only if each vertical line
intersects it in at most one point.
Example 3.1.5
If f ( x ) = 12 x + 6 , obtain the functions f (2 x ) and f ( x + 2) . What is f ( −2 ) ?
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STA102: Mathematics for Business and Social Sciences II
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Solution 3.1.5
f ( 2 x ) = 12 ( 2 x ) + 6 = 24 x + 6 and f ( x + 2 ) = 12 ( x + 2 ) + 6 = 12 x + 24 + 6 = 12 x + 30 .
f ( −2 ) = 12 ( −2 ) + 6 = −24 + 6 = −18
REFERENCES
(i) Haeussler, E.F., Paul, R.S. and Wood, R.J. (2008). Introductory MATHEMATICAL
ANALYSIS for Business, Economics, and the Life and Social Sciences-Chapter 3
Functions and Graphs (12th Ed.), pp 83-123.
(ii) Tan, S.T.: College Mathematics for the Managerial, Life and Social Sciences, Chapter
2: Functions and their Graphs & Chapter 3: Exponential and Logarithmic Functions,
pp 59-138
EXERCISES
Question 1
Given the following functions: f ( x ) = 2 x 2 − 3 x + 1, and g ( x ) = 5 x − 3, find
(i) f  g ( x )  , (ii) g  f ( x )  , (iii) f (− 4) , (iv) f  g ( 2 )  and (v) f ( 2 x + 1) .
Question 2
A car model costs P14500 with a gasoline engine and P15450 with a diesel engine. The
number of miles per gallon of fuel for cars with these two engines is 22 and 31, respectively.
The price of both types of fuel is P1.39/gallon. Thus the cost function of driving the gasoline
1.39 x
powered car x miles is C g ( x ) = 14500 +
and that for diesel car is
22
1.39 x
. Find the breakeven point and explain its significance.
C d ( x ) = 15450 +
31
Question 3
The retail price P ( x ) in Pula and the weekly demand x for a particular CD player are
related by P ( x ) = 115 − 4 x , 100 ≤ x ≤ 400.
i)
Obtain the price for weekly demand of 144 CD players.
Obtain the domain and the range for this function.
ii)
Question 4
A ball is dropped from the top of a building which is 1250 ft. tall. The distance d ( t ) of a ball
from the ground after t seconds is given by d ( t ) = 1250 − 16t 2 .
i)
Obtain the distance of a ball from the ground after 5 seconds.
ii)
Obtain the time taken to hit the ground.
Question 5
Draw the graphs of the following functions.
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STA102: Mathematics for Business and Social Sciences II
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(i)
f ( x ) = x 3 − 2. (ii) f ( x ) =
3x + 2
x
, x ≠ 1 (iii) f ( x ) = (4.5) ,0 ≤ x ≤ 5.
x −1
TEST AND EXAM QUESTIONS
TEST 1_2004-Special
Question 1
(a) Draw the graph of the function f ( x) =
1
for −4 ≤ x ≤ 2
2+ x
2
(b) If f ( x) = x 2 − 2 x and g ( x ) = x 4 , show that g  f ( x )  = x 2 ( x − 2 ) .
(c) Determine graphically the type of solution of the system:
2x − 3y = 6
.
−4 x + 6 y = 6
[8+6+6 Marks]
TEST 1_2005
Question 1
Sketch the graphs of the following functions and in each case state the domain and the range:
x
and (b) f ( x ) = 2 x − x 2 .
(a) f ( x ) =
x
[11+11 Marks]
TEST 1_2006
Question 1
1
(a) Sketch the graphs of the following functions: (i) f ( x ) = 1 − x and (ii)
2
2
g ( x) = −x − 4x − 5 .
(b) Define the derivative of a function, f ( x ) .
(c) Define (i) the existence of the limit of the function f ( x ) at the point x = k and (ii) the
continuity of the function g ( x ) at x = c .
(d) For each of the following functions, find the range of f ( x ) corresponding to the domain
given:
(i) f ( x ) = 3 − x for −1 ≤ x ≤ 4 and (ii) f ( x ) = x 2 − 4 x + 3 for 0 ≤ x ≤ 3 .
(e) Differentiate f ( x ) = 2 − x from first principles (use the definition).
(f) Given f ( x ) = 3 x + 3 and g ( x ) = x3 , find f  g ( x )  and g  f ( 5 )  .
[12+3+6+6+5+5 Marks]
TEST 1_2007
Question 1
2 + 1 − x ,
x ≤1

a) Let f ( x) =  1
,
x >1

1 − x
i)
Write down the domain and range of f (x).
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ii)
Find f (0) ,
f (−1) , and
f ( 2)
b) Given the following functions: f ( x ) = x 2 − x + 1, and g ( x ) = 2 x − 3, find
(i) f  g ( x )  , (ii) g  f ( x )  , (iii) f ( g (− 4)) , and (iv) g ( f (− 4))
c) Sketch the graph of each of the following polynomial functions and indicate the line of
symmetry in each case.
(i ) f ( x) = 12 x − 2 x, 2 for 0 ≤ x ≤ 6 (ii ) f ( x) = x 2 − 2 x − 3, for − 2 ≤ x ≤ 4
(5+12+8=25marks)
TEST 1_2008
Question 1
(a) Sketch the graph of the function f ( x ) = −3 − 2 − x .
[5 Marks]
2
from first principles.
[10 Marks]
x
(c) Find the derivatives of the following functions: (i) f ( x ) = cos ( 3 x − 1) + sin ( 2 x3 ) , (ii)
(b) Differentiate f ( x ) =
2
23 x − x
g ( x ) = ( e − ln x ) and (iii) h ( x ) =
.
[3+3+4 Marks]
2x − 3
(d) Find the equation of the tangent line to the graph of f ( x ) = x 4 + 2 x − 2 at the point (1,1) .
2x
−2
[4 Marks]
TEST 1_2008-Special
Question 1
a) For each of the following functions determine the x-intercept(s) and y-intercept(s)
i)
f ( x) = x 2 + x + 4 ii) f ( x) = x 2 − 4 x + 4 iii) f ( x) = ( x − 8) 2
ii)
Find the ( x, y ) coordinates of the turning point (vertex) and state the line
of symmetry in each of the functions above.
2
b) Plot the graph of h( x) = e1− x and determine its domain and range.
(9+6+5=20 marks)
Question 2.
1
a)
Let f ( x) = and g ( x) = x 2 + 1 , find
x
f  g ( x )  , (ii) g  f ( x )  , (iii) f ( g (− 4)) , and (iv) g ( f (− 4))
b) The demand for a certain product is given by D = 27.4 p −0.48 , where p is its price in Pula
and D is the demand in units of 1000.
Supposing that the supply of this product is given by S = p , where S is also in units of 1000,
find the value of p at which the market for the product is in equilibrium, namely the value of
p at which S=D.
(10+10=20marks)
TEST 1_2010
Question 1:
(a) Draw the graph of the function f ( x ) = x 2 ; − 3 ≤ x ≤ 4 , and hence find the
domain and range of f(x).
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STA102: Mathematics for Business and Social Sciences II
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(b) If f ( x ) = 2 x 2 + x − 12 and g ( x ) = e −2 x , obtain (i) f  g ( x )  and (ii) g  f ( x )  .
( c ) The cost function C(x) in Pula of manufacturing x number of units per day
( 0 < x < 100 ) is given by C ( x ) = 600 + 40 x − 0.2 x 2 . Determine the range. Obtain
the cost of manufacturing 60 units.
(6+6+6=18marks)
TEST 1_2011
Question 1 A
Sketch the graphs of the following functions:
(i ) f ( x) = − 2 x + 1
(ii ) g ( x) = − x 2 − 4 x − 5
(6+6= 12 marks)
Question 1 B
(a) Given f ( x) = 3 x + 3 and g ( x) = x 3 , find
(i) f [g (x)]
(ii) g [ f (5)] .
(b) For each of the following functions, find the domain and the range:
(i ) f ( x) = 2 x − 8
1
(ii ) h( x) =
.
x −1
(4+4+2.5+2.5=13 marks)
TEST 1_2012
1. Sketch the graph of the function f ( x) defined by;
− x if x < 0
, and also find its domain and range. (7 marks)
f ( x) = 
 x if x ≥ 0
2. Let f ( x) = x + 1 and g ( x) = x 2 + 1 , find f ( g ( x)) , g ( f (−1)) (2.5+2.5=5 marks)
3. For what values of q will the equation x 2 + qx + 3 = 0 have (i) Two real roots (ii) One real
root?
(2.5+2.5=5 marks)
4. Under ideal laboratory conditions, the number of bacteria in a culture is assumed to grow
exponentially. Suppose 10 000 bacteria are present initially in the culture and 60 000 present
2 hours later.
(i) Find the rate of growth of bacteria in this case.
(ii) How many bacteria will there be in a culture at the end of 4 hours.
(4+4=8 marks)
EXAM 2013
Question 1
a) Let f ( x) =
x
find the domain and range of
.
x
b) Given that f ( x ) = 2 x 2 − 3 x + 1 and g ( x) = 5 x − 3 , find g ( f ( x )).
(5+5=10 Marks)
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STA102: Mathematics for Business and Social Sciences II
Department of Statistics, University of Botswana