Mathematics, the term relation refers to a relationship between two sets of
information.
Think of the heights of all the students in your mathematics class. Now pair their
names with their heights. This pairing of heights and names is an example of a
relation.
The number of bacteria in a petri dish doubles every hour. The pairing of
number of bacteria and time is another example of a relation.
A function is special type of relation. All functions have three parts in common: the
input, the relation and the output. These three components are shown in the diagram
in Figure 3.
Let's look at the function "multiple by 3". When you put 2 into the function, the
output is
as shown in Figure 2.
We give the function a name, say f, and write it as follows:
Some functions, such as the trigonometric functions, are used so often that they
have special names; you are, for example, familiar with the sine
.
Sometimes we omit the brackets around the argument in the function name, e.g.
we often write
instead of
We select the input to a function from a set called the domain of the function. The
output of the function is called the range of the function. We'll return to the domain
and range of a function at a later stage.
Thus, a function is a rule linking one set of values (the domain) to another set of
values (the range).
A function can link a value from the domain to only one value in the range of
the function.
If a value in the domain is linked to more than one value in the range, the rule
does NOT represent a function; it is just a relation.
Are all relations functions? The last bullet suggested an answer and we'll return to this
question in more detail a little later. We first need to revise some concepts.
variable is something you are trying to measure. It may be objects, events, time
periods or even sets of numbers.
Examples
If you want to study the effect of fertilizer on the growth of plants, the
variables are plant height and type of fertilizer.
You may want to study the role of hours spent studying on your test mark.
The variables are hours studied and the test scores.
In an experiment with electric circuits the symbol i represents the current
at time t; the variables are time and current.
We distinguish between independent and dependent variables.
Independent variable: The variable which doesn't depend on another variable
In experiments, we change the independent variable and observe the
change in the other variables.
In most applications involving time, t, time is the independent variable.
Dependent variable: The variable depends on another variable
In an experiment, we change the independent variable on purpose. Then
we observe the effect of this change on the dependent variable.
Your height changed over time. Thus, your age (time) is the independent
variable since you have no control over it. As you grew up, your height
increases height depends on age (time) and is the dependent variable.
When you sketch a graph of your experimental results, the independent variable is
always placed on the horizontal axis.
When working with functions you will quite often hear the term "the argument of the
function". Is there a difference between a variable and an argument?
Argument of a function: The argument is a specific input to the function and
may be a number, a variable, a term or an expression
Activity 1
Compare "variable" and "argument".
SOLUTION
Function
Variable
3
x
x
cos 3 y
y
x 1
x
2
x
Argument
x
3y
x 1
is the difference between the domain and
the range of a function? Figure 3 depicts the
difference graphically.
Domain: What can go into a function
Co-domain: What may possibly come
out of a function
Range: What actually comes out of a
function
In this module we'll concentrate on the domain
and range of a function.
Consider the function defined by f ( x)
3x where x is a natural number.
Domain: All the natural numbers,
Co-domain: The natural numbers,
Range: Multiples of 3,
How do we determine the domain and range of a function?
The domain: Determine all the values allowed. Keep the following in mind:
The denominator of a fraction cannot be zero.
The number under a square root sign cannot be negative.
The range: Determine all the values resulting from the input to the function.
There are no fixed rules to determine the range.
For a rational function, we solve for x and then see which values of y
are not allowed.
Activity 2
Determine the domain and range of the function defined by y
SOLUTION
Graphically
The curve in Figure 4 depicts the
function y
x 3.
Note the at the point (3;0). This
indicates that the domain starts from
this point.
x 3.
From the graph we see the possible xvalues are
As a result of this
input, the possible y-values are
We don't always have a graph to help us
determine the domain and range of a
function.
Mathematically
The domain: We cannot have a negative number under the root sign.
Therefore,
and hence
The domain is the set of real values of x such that
that is,
.
The range: When
For all other values of x in the domain,
The range is the set of all real values such that
.
Before we look at another example, do you
remember the definition of an "asymptote"?
An asymptote is a line a curve approaches as
it approaches infinity. You are familiar with
the asymptotes of the tan function. The
dashed lines in Figure 5 depicts the
asymptotes x
2 and x
2 of the
function y tan x on the interval [ ; ] .
that is,
The asymptotes are not always vertical or horizontal lines. Figure 6 shows the graph
x2 2 x 1
of g ( x)
on the interval [ 2;15] . The dashed line depicts the asymptote
x 2
y x 4 of g ( x) .
Activity 3
Determine the domain and range of the function defined by f ( x)
SOLUTION
Mathematically
Since the denominator cannot be zero,
The domain:
.
Let y
1
x 3
.
that is,
1
and solve for x:
x 3
y ( x 3) 1
x 3
x
The range:
1
y
1
3
y
.
some other functions you will need more sophisticated methods which are
beyond the scope of this course.
Use the graphs to determine the domain and range in each of the cases.
1.1
1.2
2.
Determine the domain and range of each of the following functions.
2.1
f ( x) x2 4
2.2
f (t )
2 t
x
2.3
g ( x)
2.4
g (t ) 3t t 2
x 1
3.
Determine only the domain of each of the following functions.
2t 1
3.1
3.2
g (t ) 2
f ( x)
3 2x
t 1
t 5
q( s )
s 2 2s 8
3.3
p (t )
3.4
2t 3
1
3x 2
v (t )
3.5
3.6
a( x)
6x 2 x 2
t 1
question was asked earlier and the answer is a definite NO! Not all relations are
functions! Even well-known relations such as the circle represented by x 2 y 2 4 is
not a function.
A function relates every element in the domain to exactly one element in the range. A
relation that links two elements in the domain to the same element in the range, is a
function. What if one element in the domain relates to two elements in the range?
Then the relation is NOT a function!
Based on the kind of relation we distinguish two types of functions.
A many-to-one (Many one) function
More than one value from the domain linked to the
same value in the range as shown in Figure 7.
Examples: y x 2 and y sin x
A one-to-one (One one) function
Every value from the domain linked to only one value
in the range
Example: y x 3
There is no such thing as ...
many-to-many functions
All the values in the domain are linked to more than one value in the
range, and vice versa
one-to-many functions
One value in the domain is linked to more than one value in the
range
a graph represent a relation a function? Do the so-called vertical line test.
VERTICAL LINE TEST
Draw a vertical line through the curve. If it intercepts the curve once everywhere, the
relationship is a function.
See the two examples in Figure 9. The graph on the left represents a function because
any vertical line will intercept the curve once only. The graph on the right is NOT a
function; note how the vertical line intercepts the curve four times.
Is the function a one-to-one or a many-to-one function? We use the so-called
horizontal line test.
HORIZONTAL LINE TEST
Draw a horizontal line through the curve. If it intercepts the curve once everywhere,
the function is a one-to-one function.
The two examples in Figure 10 illustrates the application of the horizontal line test.
The graph on the left represents a one-to-one function because any horizontal line will
intercept the curve once only. The graph on the right depicts a many-to-one function;
note how the horizontal line intercepts the curve twice.
Write down the domain and range for each of the following sets. Then use
your answer to determine whether the set represents a one-to-one or many-toone function or neither.
1.1
1.2
2.
Determine which of the following graphs represent a function. If it is a
function, classify it as a one-to-one or many-to-one function.
2.1
2.2
2.3
2.4
are various types of classifications of functions. In the previous section, we
classify a function as one-to-one or many-to-one based on the type of relation
between the domain and range. There are several other possible ways of classification
but in this section we'll look at concepts you are already familiar with.
Polynomials: P( x) an xn an 1x n 1
a2 x2 a1 x a0 ; an 0
The coefficients a0 , a1 , a2 , , an are constants.
a n is the leading coefficient.
The powers of x are non-negative integers.
The domain of a polynomial is the real numbers.
A multinomial is a polynomial with more than one term.
Monomial: A polynomial such as 5x 3 with only one term.
Binomial: A polynomial with two terms, e.g. 2x 1 .
Trinomial: A polynomial with three terms, e.g. x 2 3 x 7 .
The degree of the polynomial is determined by the highest power.
Linear: P ( x ) mx c
Quadratic: P( x) ax 2 bx c
Cubic: P( x) ax3 bx 2 cx d
Examples
g ( x) 3x4 7 x 2 is a fourth-order polynomial.
f ( x) x2 3x 3 is a quadratic polynomial.
r ( x) 2 x 2 3x 1 is not a polynomial because of the x 1 .
p(x)
2x3
x is not a polynomial because of the root sign.
is not a polynomial because of the trig function
P( x)
where P( x ) and Q ( x) are polynomials.
Q( x)
Zeroes: The value(s) of x for which P ( x) 0 .
Poles: The value(s) of x for which Q( x) 0 .
NOTE: The function is undefined at the poles!
Rational functions: R( x)
Examples
y 1
is a rational function.
y 3y 2
3t 1
q (t )
is not a rational function because the denominator is
t2 9
not a polynomial.
R ( y)
2
Algebraic functions: Any function that can be constructed using the algebraic
operations (addition, subtraction, multiplication, division, powers and roots).
Transcendental functions: Any function that is not algebraic such as
Exponential functions
Logarithmic functions
Hyperbolic functions
Trigonometric functions
We'll look into these transcendental functions in great detail later.
Which of the following functions are polynomials? Explain why.
1.1
f ( x) x 2 3x 1 7
1.2
g ( x) 2 x x 2
1.3
2.
h(t )
2t 3 7t 2 5
1.4
i (t ) 3t 2 sin t
Are the following examples of rational functions? Motivate your answer.
x 2 3x 4
2x 1
2.1
f ( x)
2.2
g ( x)
2x 1
2x 7
t
3
2 y2 4 y 1
2.3
h(t ) 2
2.4
R ( y)
t 7t 5
y 2
transformation is a general term for one of four ways we use in Mathematics to
manipulate an object. When we apply a transformation to the graph of a given
function, we may obtain the graph of another related function think back to the
graphs of the trigonometric functions you studied at school!
How are the four categories of transformations applied to the graphs of functions?
The examples given here is for one type of transformation at a time only. We may use
a combination of a number of transformations to obtain the new graph!
graph of the faction is rotated through a specified angle to get the new function.
The rotation of graphs of functions is beyond the scope of this course.
the graph of the function about the line of reflection to obtain the new function
When you fold a paper with a graph and its reflection on the line of reflection,
you will see only one curve
Trace the following onto a piece of paper and observe the symmetry about
the line of reflection
We say the reflection is symmetric about the line of reflection.
General examples:
1.
The inverse of a function is the reflection of the function about the line
y x (more on this in a later unit).
y
f ( x ) is a reflection of y f ( x) about the x-axis, that is, the two
2.
functions are symmetric about the x-axis.
y f ( x ) is a reflection of y f ( x) about the y-axis, that is, the two
3.
functions are symmetric about the y-axis.
the graph of the function as a whole from one point to another to obtain the
new function.
If c
0:
y
y
y
y
f ( x)
f ( x)
f (x
f (x
c
c
c)
c)
shifts the graph of
shifts the graph of
shifts the graph of
shifts the graph of
y
y
y
y
f ( x)
f ( x)
f ( x)
f ( x)
a distance c upward.
a distance c downward.
a distance c to the left.
a distance c to the right.
Study the four examples below
graph of the function is stretched or compressed vertically or horizontally.
If c 0 :
y c f ( x) stretches the graph of y f ( x) vertically by a factor c.
y 1c f ( x) compresses the graph of y f ( x ) vertically by a factor c.
y f (cx) compresses the graph of y f ( x ) horizontally by a factor c.
y f ( x / c ) stretches the graph of y f ( x ) horizontally by a factor c.
Study the examples below.
Reflect the following curves as indicated.
1.1
2.
f ( x ) x3 7 x 2 5 x 1
About the x-axis
1.2
f ( x) x 2
About the line y
x
Use the given graph of f ( x) to sketch the new function g ( x) on the same
system of axes.
2.1
3.
g ( x) ( x 1) 2 3
2.2
g ( x)
x 1 2
Use the given graph of f ( x ) to sketch the new function g ( x ) on the same
system of axes.
3.1
g ( x) sin x 1
3.2
g ( x) x 3
functions are symmetrical about the vertical axis, some are symmetrical about
the origin (0;0) and others are not symmetrical at all. Based on its symmetry a
function may be classified as even, odd or neither.
Even function: The graph of the function is symmetrical about the vertical
axis (y-axis)
Mathematically: f ( x ) f ( x)
Example: f ( x ) x 2 is an even function because
f ( x) ( x) 2 x 2 f ( x )
Odd function: The graph of the function is symmetrical about the origin (0;0)
Mathematically: f ( x )
f ( x)
Example: f ( x ) x 3 is an even function because
f ( x) ( x)3
x3
f ( x)
Neither even nor odd: All functions that are neither even nor odd
The table below summarizes the theory.
Mathematically
Graphically
Examples
Even function
f ( x) f ( x )
f ( x)
x 2 , g ( x) cos x
Odd function
f ( x)
f (x)
f ( x)
Neither
No general rule
x3 , g ( x) sin x
f ( x)
x3 2 x 2
Each sketch shows part of the graph of f ( x) . Complete the graph as indicated.
1.1
Expand as an even function
1.2
Expand as an odd function
2.
Classify each of the following functions as even, odd or neither. Justify your
answer with relevant calculations.
2.1
f ( x) x5 x3 x
2.2
g (t ) (t 1) 2
2.3
h( y ) 2 y
2.4
p ( x) tan x
2.5
g (t )
t4
t
3
1
2.6
h( x )
x
1 x2
official definition of a continuous function can be quite complicated. It is
sufficient for this course to know a continuous function is a function whose graph is
a single, unbroken curve. You can thus sketch the graph without lifting your pen from
the paper. If a function is not continuous, it is said to be discontinuous.
You are already familiar with examples of continuous and discontinuous functions
think trig!
x
another look at the graph of tan x : The function "jumps" from
to
.
Thus,
the
function
is
discontinuous
at
that
point.
But
the
function
is
2
at
continuous on the interval
.
There are more functions which are continuous "in pieces" and hence we call them
piecewise continuous functions. We write the function then over several lines.
Consider the function in the sketch:
f ( x ) 3 from for x less than -1
f ( x)
x 2 3 from -1 to 1
f ( x ) x for x greater than 1
Mathematically, we write this piecewise
defined function as
3,
x
1
f ( x)
x 2 3, 1 x 1
x,
x 1
What "happens" at, for example, x 1 ? The function changes from y 2 just before
the point to y 1 just after the point. In this course it is suffice to know about the
jump; the theory of limits may be covered in more advanced courses should you
decide to continue with your education.
1.
3.
the graph of each of the following functions.
x2 , 2 x 2
f ( x)
g ( x)
2.
4,
x 2
p ( x)
3, x 1
x, x 1
x,
x,
4,
4.
q (t )
2
x ,
4,
x 0
x 0
x
2
2
x
2
x 2
function is periodic if it repeats itself at regular intervals. If a function is not
periodic, it is aperiodic or non-periodic.
Cycle: One complete repetition
Period: The smallest possible length
of one cycle
Symbol: T
Frequency: Number of cycles per
second
Symbol: f
f 1
T
Amplitude: For a wave-like function the amplitude is the maximum distance
from the equilibrium
Not all periodic functions has an amplitude! Just think of the tan function!
f ( x a) f ( x) means f is a function with period a
Here are some examples of periodic functions.
Why are the following NOT periodic functions?
1.1
1.2
2.
Each graph depicts one cycle of a periodic function. Draw two more cycles of
the function.
2.1
2.2
consider the following function:
f ( x)
x 1.
Let's revise a few concepts.
1. What is the independent variable in this function?
The independent variable is x since it is the unknown that depends on
no other variable.
2. What is the argument in this example?
The argument is x 1, the specific input for this function.
3. Calculate f (5) .
Replace the x with 5 in the function: f (5)
4. Write down f ( a ) .
Replace x with a in the function: f (a)
5. Determine f ( x 1) .
5 1
4
2.
a 1.
Replace x with x 1 in the function: f ( x 1)
( x 1) 1
x.
In the answer of the last question above, we replaced the x in the definition of the
function f with another function, i.e. x 1. We thus have one function inside another
function and we call the result a composite function.
the composition of two functions, the output of one function is used as the input to
another function. The output of the composition of the two functions is called a
composite function.
We sometimes say one function g ( x ) is inside another function f ( x ) . Thus, g ( x) is
the inside function and f ( x ) is the outside function.
Let g ( x )
g ( x) is
x 1 . Then, in #5 in the previous section, the composition of f ( x) and
x.
combination of f and g is written as
, read as "f of g of x".
Thus:
, that is, g is inside f.
, that is, f is inside g.
The symbol between the f and g is a small circle ; it is not a point as f g
means "multiply".
We sometimes use and abbreviated notation. For example, f g implies
.
The x in
is the argument of the composition. It does NOT mean
f g times x.
Does the order of f and g matter? Let's look an example to determine the answer.
Activity 1
Is ( f g )( x ) ( g f )( x ) for all functions? Use the functions f ( x )
g ( x) x 7 to motivate your answer.
SOLUTION
Determine
and
.
g
g inside f
replace every x in f with ( x 7)
g f
f inside g
replace every x in g with (2 x 3)
f
2 x 3 and
Conclusion: In general, ( f g )( x ) ( g f )( x ) .
The symbol mathematical symbol
equal sign!
is read as "implies that". It is NOT an
In Activity 1 the domain of f and g was the set of real numbers. The domain of the
composite functions
and
is also the set of real numbers. Will it
always be this easy to determine the domain of the composite function? No!
must keep the domain of both functions in mind when we determine the domain
of the composition of functions. The domain of the composition is always the smaller
of the separate function.
Activity 2
x 2 and g ( x) x2 .
Determine the domain of
if f ( x )
SOLUTION
Domain of f :
Domain of g:
The domains are the same.
Domain of f g :
3
Why does
SOLUTION
Domain of f:
not exist when f ( x) 4 x2 and g
x 3?
Domain of g:
Domain of g is the smallest.
Domain of f g :
Hence,
is not in the domain of the composite function and hence
doesn't exist.
. It is thus possible to calculate the value:
but it is the wrong answer! -4 is NOT in the
domain of f g and can thus not be used as input.
determine the composition of two functions f
1.
2.
3.
4.
f ( x ) and g
g ( x) :
Rewrite the composition, e.g.
.
Replace every x in the outside function f with the inside function g.
Simplify the answer.
Determine the domain of the composition only when asked.
Activity 4
Determine
answer.
if p
2 x 3 and q( x)
3x 2 1 . Include the domain in your
SOLUTION
Determine
and
functions.
f ( x) 3 4 x; g ( x)
1.1
2.
1.2
f ( x) 2x 7; g ( x)
1.3
1.4
f ( x)
f ( x)
for each of the following sets of
x2 1
x
x2 1; g( x)
x 2 3x; g ( x)
x 1
x 1
Determine the composite function indicated.
2.1
when f ( x) x2 3x 2 and g ( x ) 2 x 1 .
2.2
when f (t ) 9 2t and g (t )
2.3
when f ( x)
2.4
t 2 2t 3 .
x3 2 and g ( x)
t 2 and q(t ) t 2
when p(t )
3
x 2.
2t 4 .
3.
Determine
if f ( x)
x and g ( x)
domain of the composition in your answer.
x 2 2 x 1 . Include the
4.
Consider the functions p(t ) 9 t 2 and q(t )
1
.
2t
5.
6.
4.1
Calculate
4.2
4.3
4.4
Determine
.
Determine an expression for p p .
Is q q q 2 ? Motivate your answer.
Given: h( x)
.
x2 1 and g ( x)
x.
5.1
Determine
.
5.2
5.3
Determine
.
What do you notice from the results of 5.1 and 5.2?
Consider the two functions f ( x)
6.1
Determine
.
6.2
Determine
.
6
3 x
and g ( x) 3
6
.
x
6.3
What do you notice about the results from 6.1 and 6.2?
The function y
x is called the identity function.
Your conclusion in the last question will be handy in the study unit on inverse
functions.
is the "opposite" of each of the following actions?
Turn on the light.
Switch off the light.
Turn left.
Turn right.
Increase the petrol price by 23 cents.
Decrease the petrol price by 23 cents.
Each of these everyday actions has an "opposite" or "inverse" action. You are also
familiar of the "inverse" concepts when working with fractions.
Activity 1
What is the opposite or "inverse" of each of the following?
1
4
Inverse:
4
1 2
Inverse:
9
9
2
1 5
Inverse:
3
3
5
1
Inverse:
x
1
x
In Mathematics, the quantity we obtain by dividing 1 (one) by a number is quite often
called the reciprocal or multiplicative inverse.
Activity 2
Write down the reciprocal of each of the following.
1 7
Reciprocal:
2
2
7
11
1 5
2
Reciprocal: 2 15
11
5
11
5
1 y
Reciprocal:
x
x
y
1
x 1
Reciprocal:
x
x
( x 1)
Let's revisit an exponential law before we continue with the "inverse" concept.
x 1
You are familiar with the following exponential law:
1
a1
; a 0
a
Activity 3
Simplify each of the following expressions.
1
31
3
1
x x1 x
x
1
4
41
Note that a
later!
1
means "the reciprocal of a ". Keep this in mind
you'll need it
the function f ( x) x 3 . This function adds 3 to every input. The function
"subtract 3 from every input" may be written as g ( x) x 3 . Since "add" and
"subtract" are "opposite" actions, we say g ( x) is the inverse of f ( x ) .
Activity 4
Consider the functions f ( x) x 3 and g ( x ) x 3 .
a)
Sketch the graphs of f ( x ) , g ( x ) and the line y
x on one system of axes.
b)
Calculate
and
.
SOLUTION
a)
The graphs: f and g are two straight lines with the same slope and
different y-intercepts.
f g ( x)
f g ( x)
f ( x 3)
( x 3) 3
x
Let's look at p( x ) 3 x : Multiply every input by 3. The "inverse" of "multiply" is
x
"divide". Thus, the inverse of p( x) is the function q( x )
.
3
Activity 5
x
.
3
Sketch the graphs of p ( x ) , q ( x) and the line y
Consider the functions p ( x ) 3 x and q( x)
a)
x on one system of axes.
b)
Calculate
and
.
SOLUTION
a)
The graphs: Both graphs pass through the origin but have different
slopes.
p q ( x)
p q( x)
f (x)
3
x
3
x
Have another look at the two activities above.
The composition of each function and its inverse yields the identity function
y x.
The graphs of each function and its inverse is symmetrical about the line
y x.
We'll now use those facts to formalize the definition of the inverse of a function.
DEFINITION
Let f f ( x ) and g g ( x ) be two functions. If
then g is the inverse of f, and f is the inverse of g.
We have to calculate
inverse functions.
and
NOTATION
The inverse of the function f
to prove that f and g are
1
.
f ( x)
f 1 ( x) . Here is an exception: If f ( x )
FACTS TO REMEMBER
1. Mathematically: f f
2. Graphically: f and f
,
f ( x) is written as f 1 ( x) and read as "f inverse of x".
Very important! f 1 ( x)
Usually, f ( x)
AND
1
1
1
, then f 1 ( x)
x
f 1 f x.
are symmetrical about the line y
3. The inverse of an inverse is the original function:
1
.
x
x.
.
The range of f f ( x ) becomes the domain of f 1 as depicted in the diagram in
Figure 1. Thus, to calculate the inverse of a function, we may swap x and y in
y f ( x ) so that x f ( y ) and then solve the "new" y to obtain the expresiion for the
inverse function.
STEPS TO DETERMINE THE INVERSE
1.
Let y f (x ) .
Swap x and y.
2.
3.
Solve for y.
4.
Check for limitations on x (the domain).
5.
Write down f 1 (x) .
Write down the domain of the inverse.
6.
Activity 6
Determine the inverse of the function defined by f ( x )
SOLUTION
Let y 2 x 5 .
Inverse
Swap x and y:
x 2y 5
Solve for y:
x 5 2y
x 5
y
2
This function is valid for all x
x 5
f 1 ( x)
2
Activity 7
Determine the inverse of f ( x)
SOLUTION
Let y x3 1
Inverse
x y3 1
x 1
2x 5 .
x3 1.
y3
y 3x 1
This function is valid for all x
f 1 ( x) 3 x 1
The graph in Figure 2 depicts f ( x)
Note the symmetry about y x .
x3 1 and its inverse.
Activity 8
Determine the inverse of p(t )
t
t 1
; t
1.
Be careful! There is no x in the function!
SOLUTION
t
Let y
t 1
Inverse
y
t
t ( y 1)
y 1
y
ty t
t
y
y ty
y (1 t )
t
; t 1
1 t
y
p 1 (t )
t
1 t
; t 1
Is the inverse of a function always a function?
do another activity to answer this question.
Activity 9
Consider the function f ( x) x 2 .
a)
Calculate f 1 ( x ) .
b)
Sketch the graph of f and f 1 on one system of axes.
c)
Is f 1 ( x) a function? Motivate your answer.
SOLUTION
a)
Let y x2 1 .
Inverse
x y2 1
x 1 y2
y
x 1
But the square root of a negative number is not a real number. Hence
x 1 0 for the inverse and thus x 1.
f 1 ( x)
the
b)
x 1; x 1
in front of the root sign.
The graph of f and its inverse
c)
Perform the vertical line test on the graph of
f 1 ( x) : The line intercepts the curve twice.
Hence, the inverse is NOT a function.
Perform the horizontal line test on the graph of f ( x) in the activity above: A
horizontal line intercepts the graph more than once. Thus, f ( x) is a many-to-one
function.
one
functions have inverse functions!
function: The inverse of the function exists and is a function.
APPLICATION 1
You've used inverse functions in the past without, maybe, realizing it!
You now have the knowledge: Quickly sketch the graph of
for
Then sketch the graph of its inverse on the same system of
axes. Is the inverse a function?
Activity 10
Solve for x if sin x 0.6 .
SOLUTION
You probably used the
answer:
x 36.9
key on your clever calculator to obtain the
The function sin 1 is the inverse of the familiar sine function. We'll return to the
inverse trigonometry functions in a later study unit.
APPLICATION 2
The graphs in Figure 3 depicts the function
y 2 x and its inverse.
We'll revisit those functions in a later study unit
and show that one is the inverse of the other.
Sketch the inverse of the given function on the same set of axes. The dashed
line represents the graph of y x .
1.1
1.2
1.3
2.
1.4
Determine the inverse of each of the following functions.
x 5
f ( x) 3 x 7
g ( x)
2.1
2.2
3
3x 2
3
h( x )
2.4
p( x )
2.3
5
x 3
t 1
2.5
q (t ) (t 1)3
2.6
k (t ) 3
2
t
2.7
f ( x) 3 2 x
2.8
f (t )
,t
t 4
3
2.9
2.10
f ( y) 3 y 7
f (s) 2 2 , s 0
s
x 1
2.11 u ( x )
3 2x
2.12 y
,x
2x 1
4
1
2
3.
Are the following sets of functions inverses? Use the definition of inverse
functions to justify your answer.
2 x
x
f ( x)
; g ( x)
3.1
x
2 x
t 1
3.2
p(t ) 5
; q(t ) 2t 5 1
2
4.
Consider the function defined by f ( x)
Motivate your answer.
5.
Determine the inverse of each of the following functions. Clearly state the
domain and range of the function and its inverse.
5.1
5.2
f (t )
t 2
g (t )
t 2
6.
Given: f ( x)
2x 4 and g ( x) x2
6.1
Determine f g and g f .
2
x2 2 . Is this function invertible?
2.
6.2
Is g the inverse of f ? Why?
7.
Use the function f, represented by f ( x) 3x 2 , to explain the following
statement: "A function and its inverse is symmetrical about the line y x ."
8.
Is the following function invertible? Motivate your answer.
f ( x) x3 4 x 2 3x
"far" is 4 from 0 on the number line? What is
the distance between 0 and -4 on the number line?
The answer to both questions is 4, as indicated in
Figure 1.
In general, the "distance" from a number to 0 on a number line is called the "absolute
value" of the number. Here are a few examples:
The absolute value of 4 is 4.
The absolute value of -4 is 4.
The absolute value of 7 is 7.
The absolute value of -3 is 3.
The absolute value of -101 is 101.
Thus, the absolute value of a number is always the value of the number without its
sign.
DEFINITION
The absolute value or modulus of a real number a, denoted by a , is the numerical
value of the number without its sign.
More examples:
2 2 and
7.6
2
7.6 and
2
7.6
7.6
and
2
2 but
2
2
Simplify the following expressions.
6 6
1.1
2.
6 6
1.2
1.3
3 4
1.4
3
4
1.5
4 3
1.6
4
3
1.7
2 6
1.8
2 ( 6)
Justify your answer in each case with relevant examples.
2.1
Is a b a b for all real values of a and b?
Is a b
2.2
know that 3
question.
Activity 1
Solve for x if x
3 and
a
b a, b
3
?
3 . Keep this in mind and answer the following
5.
SOLUTION
Using intuition: Based on the example above,
x 5 because 5 5 and x 5 because
Mathematically:
x 5 or
x 5
x
5
5
5
The sketch on the left illustrates the solution of
x 5 : The value of a number x that is five units
from 0. Why is the distance measured from 0? Because x
x 0
5!
How will we determine the solution of x 1 3 graphically?
The left-hand side of the equation has a 1 instead of a 0.
Look for all values 3 units from 1.
Activity 2
Solve for x if x 1
SOLUTION
x 1 3
x
x
3.
or
x 1
x
4
3
2
2 or x 4
In general, the solution of ax b
( ax b) c .
Activity 3
Solve for x if 2 x 1
5.
c is obtained by solving for x in ax b c and in
SOLUTION
2x 1 5
2x 4
x 2
x
or
(2 x 1) 5
2x 1 5
2x
x
6
3
3 or x 2
An absolute value equation must always be in the standard form, which means
there must be a positive value on the right-hand side of the equation.
Why does the solution of x 1
negative!
1.
the following equations.
2x 3 7
2 not exist? The right-hand side is
2.
t 1 3
3 n
3.
2a 1
7
4.
5.
6 2x
24
6.
7.
3x 2
4
8.
2
1 x
3
2
4x 2 8 0
inequality implies that two values are not equal.
x y : x is not equal to y
x y : x is less than y
x y : x is less than y
x y : x is less than or equal to y
x y : x is greater than or equal toy
Reminder 1: When you divide by a negative number in an inequality, remember to
change the inequality!
If x 3 , then x
3
Reminder 2: 2
x 5 means x (2;5) and 2 x 5 means x [2;5] .
Reminder 3: Expressions such as 2 x 3 and 2
Activity 4
Solve for x if x 2
3.
x 3 has no meaning.
SOLUTION
x 2 3
x 5
and
( x 2)
x 2
3
3
1
x
1 x 5
5
Solve for x if 3 x
SOLUTION
3 x 2
x
1
x 1
2.
or
(3 x ) 2
3 x 2
x
5
x 1 or x 5
the following inequalities.
2.
t
2
2x 5 1
4.
5 4x
5.
t
3
2
6.
3x 1 4 1
7.
3 4x 1
8.
5
1.
2x
3.
6
10
9
3
1
x 1
9
now know that 2 2 and 2 2 . Based on this fact, we define a new function,
called the modulus function as follows.
The modulus function is defined as f ( x )
x
x
if
x 0
x if
x 0
for all real
values of x.
This is an example of a piecewise defined function because the function is
defined in two "pieces": y x on the interval x [0; ) and y
interval x ( ;0) a shown in Figure 2.
x on the
Figure 15 The modules function
Sometimes the modulus function is called the absolute value function.
Domain: x
; Range: y [0; ) .
The modulus function in its basic form f ( x)
function.
x is an even, aperiodic
basic modulus function is given by f ( x) x . We may "change" the graph of a
modulus function by, for example, flipping it about the x-axis or moving it up by one
unit as shown in Figure 3
The sketch in Error! Reference source not found. compares the graphs of
y x 2 1 and y x . The minimum point (2;1) of y x 2 1 is called the
salient point or vertex.
The general or standard form of a modulus function is
f ( x) ax b c .
Note the negative sign in the general formula!
The coordinates of the vertex are given by
The graph is symmetrical about the line x
.
b
.
a
STEPS TO SKETCH THE GRAPH
1. Write the function in the standard form f ( x) ax b c .
2. Write down the coordinates of the vertex.
3. Draw the line of symmetry through the vertex.
4. Calculate the y-intercept, if any.
5. Calculate the x-intercepts, if any.
6. Sketch the graph by connecting the points and using symmetry.
Activity 6
Sketch the graph of f ( x)
x 1 3.
SOLUTION
NOTE: a 1; b 1; c 3
Vertex: ( x; y ) (1;3)
Line of symmetry: x 1
y-intercept:
y 0 1 3 1 3 2
x-intercept:
( x 1) 3 0 or ( x 1) 3 0
x 4
x
2
Connect the points.
7
Sketch the graph of g ( x) 2 x 1 1 .
SOLUTION
1; c 1
NOTE: a 2; b
1
Vertex: ( x; y) ( 2 ;1)
1
Line of symmetry: x
2
y-intercept:
y 1 1 2
x-intercepts:
2x 1 1 0
2x 1
1
No solution
No x-intercepts
Connect the vertex and the yintercept, and use symmetry to complete the graph
Sketch the graph of each of the following functions.
1.1
f ( x) 1 x
1.2
g ( x) 1 x
1.3
2.
p(t )
t 2 3
1.4
q(t )
2t 3 4
Consider the function f ( x) x 1 3 .
2.1
Calculate f (5) and f ( 5) .
2.2
Determine the intercepts of f ( x ) with the axes.
2.3
Write down the coordinates of the salient point.
2.4
Hence sketch the graph of f ( x ) on the interval
.