AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

Mathematic pure 1 (functions) email:racsostudenthelp@gmail.com
2. FUNCTIONS
(I) Function: A function is a special relationship between values: Each of its input values
gives back exactly one output value. It is often written as "f(x)" where x is the value you
give it.
(http://tutorial.math.lamar.edu/Classes/Alg/FunctionDefn.aspx)
Example: y = 4x - 3. ---- (i)
 if we are to find the value of y, when x=2, then we would substitute as
follows,
y = 4(2) - 3
y = 5
Alternatively, equation (i) above could be written as follows,
F(x) = 4x - 3-(ii)
 f(x) is an alternative way of expressing y, hence equation (i) is exactly
the same as equation (ii)
f(x) = 4x - 3, ---- (ii) , in this case if I am to substitute x=2 in
the equation, it would look as follows:
f(2) = 4(2) - 3
f(2) = 5.
 For a better visual understanding of what functons are , consider the below
diagram.
RACSO PRODUCTS Page 1

y=f(x)
The equation of the graph above is f(x) = x +2.
From the definition of a function above “A function is a special relationship between values:
Each of its input values gives back exactly one output value”
-when
x = (1) f(1) = (1) + 2 = 3
X = (2) f(2) = (2) + 2 = 4
X =(-1) f(-1) = (2) + 2 = 4
X =(-2) f(-2) = (2) + 2 = 4
- The output values are the y-values or f(x)-values , if you go to graph above and
choose any point on the x-axis, if you plug it in the equation it should give you the
corresponding y-value/ f(x).

Extra information( not necessary for this subject)
 You might have come across the word, independent variable and dependent
variable
 The y-value is the dependent variable( depends of x-value) , As you can not get
the value of y/ f(x) unless you plug in the value of x to the equation. Therefore, y
is the dependant and x is independent( does not depend).
Something to note,
If x= 2+t, and equation given is f(x)= x +2, don’t be confused, since
x= (2+t) plug in the whole thing, and therefore the function will look like f(2+t)
= (2 +t) +2 = 4+t
f(x)= x + 2
f(2+t) = (2 +t) + 2 when, x=2+t.
(II) Domain and range.
What is a domain? What is a range? Why do I need to learn this?
Domain is the set of all first elements of ordered pairs (x-coordinates).
Range is the set of all second elements of ordered pairs (y-coordinates).
(http://www.regentsprep.org/Regents/math/algtrig/ATP5/DomainRange.htm)

Example 1 consider the linear graphs below, they have same equation f(x) =x
Graph (a) ( -4) (4) Graph (b)
For graph(a) For graph(b)
Domain: x ε all real numbers Domain: -4 ≤ x ≤ 4
Range : y ε all real numbers Range : -4 ≤ x ≤ 4
REMEMBER -The domain and range states to what point the graph goes to both the x
and y-axies.
Notes

Remember
A line goes to infinite and has two arrows ( )
A line segment has no arrows and has a beginning and has an end.( )
Example on the graph (a) above, notice that the line has two arrows, meaning it is going to infinite.
Vertically ( range) , goes to infinite hence takes all real numbers
Horizontally (domain) goes to infinite, hence takes all real number.
Example 2
Equation of graph is f(x) = x2
Figure(c)
Domain: -3 ≤ x ≤ 3
Range : ???
Range: 0 ≤ x ≤ 3
Domain: -2.83 ≤ x ≤ 2.83

Figure(d)
Notice there are
arrows
 Notice the difference, in figure (c ) and figure ( d) - arrows
Figure (e)
. - x ≠ 0 , means x is not equal to zero
F(x) takes all
value of y≥0
x- Takes all
real number.
The equation of figure (e) is f(x) = ퟐ
풙−ퟐ
Domain: x ε all real, but x ≠ 2
Range: y = ε all real, but x ≠ 0
(ε = elements)
Domain: x = all real, but x ≠ 2
Range: y = all real, but x ≠ 0
Domain explained: Domain of the above graph is such that it takes all real numbers, except 2. If you plug 2 in the
equation f(x) = 2/ x - 2 ( the equation of figure (e) above) , the answer is undefined, but any other number works.
Range explained: the range can be easily determined by either using the equation of figure (e) or by looking at
the graph.

 How to obtain range using the equation
From the equation f(x) = 2/ x - 2, think of a value that when plugged as x it won’t work. You will realises
that there is no real number that can be plugged in as x-value to obtain f(x) = 0. Therefore range can be
any other real number except 2.
 How to obtain range from the graph above: it is clearly seen that the doesn’t not touch (intersect), does
not intersect the x-intercept, that means y/ f(x) can never be zero!
(III) One –to–one function: it is a function for which every element of the range of
the function corresponds to exactly one element of the domain.
- In exams you are always asked if a function is one to one, or how to change a graph to make it one to one.
-To obtain whether a graph is one-to-one we use, the horizontal test.
-What is horizontal test? Simply drawing horizontal line on a graph to se e if intersects the graph more than
once
Example:
Figure (f)
One-to- one function
-Linear graphs are always one-to-one
functions,
( linear means straight)
-Linear graphs are straight line graphs.
-Horizontal test worked successfully ( no line
crosses the graph more than once.
Horizontal lines
intersect graph once
Horizontal line does
not intersect graph
once
Yes
Not

When the y-value of
f(x) = 4
This graph is quadratic; it is not a one-to- one function because
the horizontal line test intersects the graph more than once .
- The equation of the graph in figure (f) , is
- f(x)= ¾ x2-3.
-In simple language, one y-value should give us one
x-value. But if we plug in f(x)= 4, you obtain 2
values of x, ( x= 3.055 or x= -3.055)
 Therefore it is not a function, because for one
value of f(x) you obtain 2 values of x.
 Note: A quadratic function be restricted by changing
the domain
 -As illustrated in figure g(i) and g(ii) on the
following page
Figure (f)
X=- 3.055 X = 3.055
Figure (g)
When f(x)= 4, there is
only one value for
x=3.055
g(i) g(ii)
When f(x)= 4, there is
only one value for
x=-3.055
Look at the graphs g(i) and g(ii) do you see the difference between figure (f) and figure g(i) and g(ii) ?
Graph g(i) has an equation is f(x) = ¾ x2- 3, x ≥ 0
Graph g(ii) has an equation is f(x) = ¾ x2- 3, x ≤ 0
Note how domain is
used to restrict the
graph.

- That means the only way to make a graph one to one is to either eliminate the
left side g(i) or the right side g(ii)
Question and answer for domain and range,
1. Are the following a one-to-one functions?
Answer: yes it is a one-to-one function because there is one element for the f(x) corresponds
to exactly one element of the x
2)
Answer: No, it is not one-to-one function because
one value of f(x) corresponds to more than one
value of x.
-Is it possible making it one –to –one function? Yes it
Is, will see that in trigonometry
It is one to one only when 1.5 < x< 5 .
This one is one to one function

(iv) Inverse function: In mathematics, an inverse function is a function that "reverses"
another function:
Example: f(x) = 2x + 3 , y= 2x + 3,
A B
1
2
3
4
5
7
9
11
2(x) + 3
Inverse function
x= 2y + 3
Y = x-3/ 2,
But, Y = f(x)-1 so
Equation is
f(x)-1 = x-3/ 2
Switch y and x
F(x)-1 = x-3/ 2
-Noticed, with the equation f(x) = 2x + 3 , it takes you from 1 ( in set A) to 5 ( in set B) and with
equation f(x)-1 = x-3/ 2 take the 5 ( in set B ) back to 1 in sets 1.
F(1) = 2(1) + 3 = 5 and f(5)-1 = (5)-3/ 2 = 1.
Make Y the subject
Note: f(x)-1 means the inverse of y, after you have switched y and x, make y the subject
and the new y becomes f(x)-1.
Questions to try ( They will be solve in my the video)
1. y = 2x – 4
2. y= 2x2 + 4
3. y= 2x2 + 4x - 4

(V) Composition of fuctions:
1. Example: f(x) = 2x – 4 h(x)= 3x - 6
When composed: i) f(g(x)) (ii) h( f(x))
i) f( g(x) ) = 2 ( h(x) ) – 4 = 2(3x – 6) – 4 = 6x - 18
f (x) = 2 x – 4
therefore: f(g(x)) = 6x - 18
-In finals exams, they may give you. f(g( 4))
-After solving to the point where f(g(x)) = 6x – 18 , plug in x=4,
-Answer = f(g(4)) = 6(4) – 18 = 24 – 18 = 6
The same idea, with (ii)
ii) h( f(x)) = 3( f(x) ) – 6 = 3(2x – 4) – 4 = 6x- 16

Illustrate in graphical terms the relation between a one-one function and its inverse.
-REMEMBER: ONLY ONE -TO ONE-FUNCTION HAVE INVERSE!
Look at the graphs below, they are not a one-to-one function.
Equation of the graph
is f(x) = 3x2
Do you remember drawing a
graph when given the equation?
1. If you draw the graph of the
inverse of f(x) = 3x2 which is
f-1(x) = √ x/3, you get the blue
graph in figure(h2) below.
Figure (h1)
-Previously we learnt that
horizontal line test determines
whether the function is a one to
one.
Vertical line test determines
whether an inverse of a graph
relation
NOTE: THE DOMAIN OF A FUNCTION IS THE RANGE OF ITS INVERSE AND VICE VERSA:

Vertical line test is only for test inverse functions.
It is done by drawing vertical lines on the inverse function graph
as shown in the figure (h2). It tells us whether the relation is a
function. If it crosses the graph more than once means relation
not a function
Equation of the graph
is f(x) = √ x/3
Figure (h2)
-Look at the two graphs carefully,
-Have you noticed that graph inverse f-1(x) = √ x/3 is the
image of f(x) = 3x2 along the dotted line y= mx?
- Assume the dotted line is the cross-section of a mirror and
line y=3x2 is an object in front of a mirror, and therefore
graph f-1(x) =√
푥
3
is the image you see in the mirror.
Note: The equation the dotted line is y=mx.
-THEREFORE THE f-1(x) IS THE REFLECTION OF F(X) ALONG THE
LINE Y=X.
Figure h3

CAMBRIDGE PAST PAPER QUESTIONS

Most of the graphs are from the link below
“ https://www.google.co.za/search?q=a+graph+showing+range+and+domain+in+word+format&tbm ”

AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

More Related Content

What's hot

Viewers also liked

Similar to AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS

More from RACSOelimu

Recently uploaded

In this document

AS LEVEL Function (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS