Topic 1: Functions and graphs – 1.1 Revision of basic concepts
(James 5th edn: §§2.2.1, 2.8.3)
Lecture 1: domain
range
graphing functions
algebra of functions
piecewise-defined functions
The learning objectives of this lecture
Most of the first few lectures should be revision material, but do not be misled into thinking that you will
have seen and understood everything that is covered in ENG1090 – even if you previously attempted
equivalent material at Year 12 (such as in VCE Specialist Mathematics, for example). In particular,
different notation and terminology may have been taught, and maybe not always the ‘whole truth’!
That said, this first lecture aims to:
Review some basic terminology, concepts and conventions that you may (or may not) have seen
before, but should understand before the unit gets underway;
Outline how to determine the domain and range of functions (using known properties of basic
functions, not only graphically);
Illustrate how to sketch simple graphs of functions (without a calculator);
Show how to form more complicated functions by combining together simpler functions
algebraically, and explain how to determine the domain of the combined function.
Review piecewise-defined functions, which use different formulae over parts of the domain.
L01 – Functions and graphs – Revision of basic concepts
Page 1
•
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ENG1090
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Basic concepts and terminology†
1.
A function is a mathematical object used to represent a special kind of relationships between two sets of
objects: for each object in the first set, it assigns uniquely an object in the second set.
More precisely:
A function f : A B is a rule that assigns to each element x in set A a unique element
y f ( x) in a set B . Here, x A is called the independent variable and y is called the
dependent variable (since the value of y depends on the value of x , as given by y f ( x) ).
It is important to know and understand the typical terminology here:
The set A is called the domain of the function and for a function f we denote that by Dom f .
Typically the domain of a function is specified as part of the definition of that function. (The set B is
sometimes called the codomain, although we will rarely need or use that terminology in ENG1090.)
For each x in Dom f , the value y f ( x) under the function f is called the image of x under f .
The set of all images of f is called the range of f , denoted by Ran f . Note that Ran f B , but
Ran f might not be all of the codomain B.
Throughout most of ENG1090 we will work with functions for which A and B are sets of real numbers.
The set of all real numbers is denoted by R (or ). Thus f : A B where A R , B R .
[ † Revision material ]
L01 – Functions and graphs – Revision of basic concepts
Page 2
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A function can be defined by a table, a graph, a mathematical formula or in words. When defining a
function, the domain should also be stated clearly. For example, for the “squaring function” we might use:
table
x
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-2
-1
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1
2
3
4
5
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5,5
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Technically, these are three different functions above, as the domain is different in each case.
The maximal (“largest possible”) domain of the squaring function is the set of real numbers R, and the
corresponding range of that function is the set of non-negative real numbers, i.e. R {0} [0, ) .
L01 – Functions and graphs – Revision of basic concepts
Page 3
School of Mathematical Sciences
ENG1090
Defining a function using a formula (or relation) and a clearly-stated domain is ‘best practice’, but …
If a function is given by a formula and no domain is specified, for ENG1090 we use the set of
all real numbers for which the mathematical expression is defined (i.e., makes sense) as the
domain of the function. This is sometimes known as the maximal domain of that function.
Recall that:
division by zero, and
square (and other even) roots of negative numbers
are not defined, so we exclude points from the assumed domain where this would occur.
For example, consider the function
a2
f ( x)
x 2 , where a 0 .
In this context, x is the variable and a is an unspecified constant (sometimes called a ‘parameter’). This
function is defined for all real numbers x such that the expression within the square root is non-negative,
which requires that a2 x2 0 . After manipulation, this is true when x satisfies x2 a 2 , or a x a
when a 0 , and hence the maximal domain of this function is
Dom f
[ a, a] {x : a
x a} .
Note: There is an important distinction between numerical constants and fixed (but unspecified)
parameters, such as a above, and variables, like x above. In introductory mathematics units most
functions involve constants, such as in 1 x 2 , but advanced units may use parameters like a instead.]
L01 – Functions and graphs – Revision of basic concepts
Page 4
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Graphs of functions†
When a function is given by a formula, it is useful to also have a graphical representation of it as that can
help us visualise the behaviour and properties of the function. In ENG1090 you will need to know how to
draw graphs of most simple functions without using your calculator, including low-degree polynomials
as well as scalings and translations of circular, exponential and logarithmic functions.
Technically, the graph of a function f (x) is the set of all points of the form ( x, f ( x)) in the coordinate
plane, where x is in the domain of f . It is expected that you already know well the graphs of some simple
and common functions, for example:
12
4
10
2
-2
1 x
-1
8
2
0
6
4
2
-4
-1
-6
1
2
f (0)
y-intercept: y g (0)
1
Zero: x 1/ 3
Zeros: x 1, x
L01 – Functions and graphs – Revision of basic concepts
Page 5
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ENG1090
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To sketch the graph of many simple (and or even complicated) functions, it can help to be aware of their
basic properties, especially features that can often be identified easily, such as:
values of x where the function is either equal to zero or undefined;
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the location of any ‘turning points’, such as local maxima and minima;
values of the function at several selected points, especially simple points like x 0 , x
1 , etc.;
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,
whether the function (consistently) is increasing or decreasing anywhere, as x increases;
the locations of any vertical or horizontal asymptotes;
other distinctive features, for example that its values repeat over a particular period.
If it is also reasonable that the function will vary continuously and smoothly over regions (or all) of the
domain, then the form of the function in between the values above can typically be inferred by carefully
sketching a smooth curve that satisfies those properties.
Sketched graphs of functions for ENG1090 do not need to be perfect or very accurate, but they should
capture the main features. And always label both of the axes, including with some typical numbers.
The same approach as above can also help determine the (maximal) domain of many given functions, as
well as to help identify its range, even without attempting to graph the function.
As an example, consider the function f ( x)
{0} [0, ) and at x 0
x . This has maximal domain R
its value is f (0)
0 0 . For x 0 it increases everywhere, and for large values of x its value increases
without limit. Also its slope f '( x) 12 x 1/2 is large for small x and decreases as x increases. Along with a
few typical values, this enables the function to be plotted, and indeed demonstrates that Ran f [0, ) .
L01 – Functions and graphs – Revision of basic concepts
Page 6
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Algebra of functions†
3.
Functions can be combined algebraically to form new functions. Provided we are careful with their
domains, we can add, subtract, multiply and divide functions in much the same way as we add, subtract,
multiply and divide real numbers.
Given two functions f and g with domains A and B respectively, the function ( f g ) is defined so that
the image for each value of x is the sum of the corresponding images of f and g, that is
(f
f ( x) g ( x) .
g )( x)
This makes sense only if x belongs both to the domain of f and to the domain of g, i.e x
hence the domain of ( f g ) is
Dom( f
g)
A
B , and
A
Dom g .
B Dom f
Similarly, the difference, product and quotient functions can be defined by
(f
g )( x)
f ( x) g ( x )
( f g )( x)
f
( x)
g
f ( x) g ( x)
f ( x)
g ( x)
where
Dom( f
where
Dom( f g ) Dom f
Dom g
where
f
Dom ( ) (Dom f
g
Dom g ) \{x : g ( x) 0}
g ) Dom f
Dom g
L01 – Functions and graphs – Revision of basic concepts
Page 7
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4.
Piecewise defined functions
These are functions defined in terms of a different formula in different parts of the domain. They often
arise in engineering applications, as in reality life is not always simple (and often simple is not useful)!
These types of functions are sometimes called hybrid functions but in ENG1090 the term piecewise
defined functions is used, or sometimes just piecewise functions.
For example, consider:
(a)
1 x
x2
f (x)
x 1
x 1
(b)
g (x)
10
x,
x,
x
x 0
x 0
4
8
3
6
2
4
1
2
0
-1
1
x 2
-4
3
-2
0
2 x
4
Often these piecewise-defined functions include jumps (known as “discontinuities”) in value or slope.
However, they need not have that property – as it may be simply that more than one type of formula is
needed to express the function over all of the x values in the domain.
L01 – Functions and graphs – Revision of basic concepts
Page 8
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