2. CONTENTS
I FUNCTIONS AND THEIR GRAPHS
1. Functions
i) Function definition
ii) The Vertical line test
iii) Absolute value of a number
iv) Symmetry
2. Essential functions
i) Linear function of x
ii) Polynomials
iii) Power function
iv) Rational function
v) Algebraic function
vi) Transcendental function
3. New functions from old functions
i) Transformation of functions
ii) Combination of function
iii) Composition of function
II TRIGONOMETRY
1. Trigonometric functions
2. Graphs of the trigonometric functions
3. Trigonometric identities
4. Solving trigonometric equations
III GRAPHS OF SECOND-DEGREE EQUATION
1. Circles
2. Parabolas
3. Ellipses
4. Hyperbolas
2
3. I FUNCTIONS AND THEIR GRAPHS
1. Functions
i) Function definition:
A function is a rule that assigns to each element in a set D exactly one
element, called in a set R.
f x
( )f x
D and R are sets of real numbers.
The set D is called the domain.
The range R of is the set of all possible values of as varies
through out the domain.
f ( )f x x
Example 1:
Sketch the graph and find the domain and range of each function
a) b)12)( −= xxf
2
)( xxg =
Solution:-
a) The graph of the equation 12 −= xy is a line
So we need any 2 points to sketch its graph.
1 1
2 2
0 1 :(
0 :
x y
y x
= ⇒ = − −
= ⇒ =
0, 1)
( ,0)
The domain of is the set of all real numbersf R
The range is also R
b) The graph of the equation is a parabola
2
)( xxg =
The domain of is Rg
The range of g is [ )0,∞
Example 2:
Find the domain of each function
a) 2)( += xxf b)
xx
xg
−
= 2
1
)(
Solution:-
a) Because the square root of negative numbers is not defined, the domain of
consists of all values of such that
f
x 2 0x+ ≥ . The domain is .)2,⎡
⎣− ∞
3
4. b) Since
)1(
11
)( 2
−
=
−
=
xxxx
xg and division by 0 is not allowed,
we see that is not defined when)(xg 10 == xorx
Thus, the domain of g is ( ) ( ) ( ),0 0,1 1,−∞ ∞∪ ∪
ii) The Vertical line test
A curve in the - plane is the graph of a function of if and only if no
vertical line intersect the curve more than once.
xy x
A function of x not a function of x
iii) Absolute value of a number
Definition: a is the distance from to 0 on the real number linea
⎩
⎨
⎧
<−
≥
=
0
0
aifa
aifa
a
Properties of absolute values:
For , then0>b
(i) ba < Iff bab <<−
(ii) ba > Iff baorba −<>
(iii) ba = Iff baorba −==
Example 3:
Sketch the graph of xxf =)(
Solution:-
⎩
⎨
⎧
<−
≥
=
0
0
xx
xifx
x
4
5. iv) Symmetry
Even function:
If a function satisfiesf )()( xfxf =− for every number in its domain,
then is called an even function.
x
f
e.g. xxfxxf cos)(,)( 2
==
2
)( xxf = An even function
Note: the graph is symmetric with respect to the y-axis
Odd function:
If a function satisfiesf )()( xfxf −=− for every number x in its domain,
then is called an odd function.f
e.g. xxfxxf sin)(,)( 3
==
3
)( xxf = , an odd function
The graph is symmetric about the origin.
Example 4:
Determine whether each of the following function is even, odd, or
neither even nor odd:
a) b) c)xxxf += 5
)(
4
1)( xxf −= 2
2)( xxxf −=
Solution:-
)()()( 55
xfxxxxxf −=+−=−−=−a) Odd function
b) Even function)(1)( 4
xfxxf =−=−
c) Neither even nor odd
2
2)( xxxf −−=−
5
6. 2. Essential functions
i) Linear function of x:
we mean that the graph of the function is a line y mx c= +
Equation of a line where ( )1 1,x y , ( )2 2,x y are 2 given points
1 2
1 2
y y y y1
1x x x x
− −
=
− −
Equation of a line, where given point and is the slope( 11 , yx ) m
( )1
1 1
1
y y
m y y m x
x x
−
= ⇒ − = −
−
x
Special lines
Two lines 1 2&L L 2have slopes are:1 &m m
(i) Parallel if 21 mm =
(ii) Perpendicular 121 −=mm
Example 5:
Find an equation of the line passes through the points A(-1, 4) and B(3, 2)
Solution:-
2
1
4
2
31
24
1
4 −
=
−
=
−−
−
=
+
−
x
y
71
2 2
y x−= +
Example 6:
Find an equation of the line passes through the point (3, -4) and is parallel to
the line 952 =− yx
Solution:-
From the given line:
2 2
2 5 9 5 2 9 9
5 5
x y y x y x m− = ⇒ = − ⇒ = − ⇒ =
∴The required equation is
m
x
y
=
−
+
3
4
⇒
5
2
3
4
=
−
+
x
y
⇒
5
26
5
2
−= xy
6
7. ii) Polynomials
A function p is called a polynomial if 0
1
1 ........)( axaxaxp n
n
n
n +++= −
−
Where is a nonnegative integer and the numbers are
constant called the coefficients of the polynomial. The domain of a
polynomial is . the degree of the polynomial is .
n naaa ,...,, 10
R n
e.g.
6 4 3
( ) 2 25 2p x x x x= + + + is a polynomial of degree 6
A polynomial of degree 1 is of the form ( )p x mx c= + and so it is a
linear function.
A polynomial of degree 2 is of the form and is
called a quadratic function. Its graph is always a parabola obtained by shifting
the parabola , the parabola opens upward if , and downward
if
cbxaxxp ++= 2
)(
2
axy = 0>a
0<a
iii) Power function ( ) af x x=
, where is a positive integera n= n
7
8. 1a
n
= , where is a positive integer:n n
xxf
1
)( = is a root function
The graph of the reciprocal function1a =− 1( )f x
x
=
iv) Rational function
A rational function is a ratio of two polynomialf
)(
)(
)(
xQ
xp
xf =
e.g.
4
12
)( 2
24
−
+−
=
x
xx
xf
v) Algebraic function
It can be constructed using algebraic operation (addition, subtraction,
multiplication, division, and taking roots)
e.g. 2( ) 1f x x= + ,
4 216( ) x xg x
x x
−=
+
vi) Transcendental function
Functions that are not algebraic. The set of transcendental function include the
trigonometric, inverse trigonometric, exponential and logarithmic function …
e.g. ( ) sinf x = x , ( ) 5xg x =
8
9. 3. New functions from old functions
i) Transformation of functions:-
By applying certain transformations to the graph of a given function we can
obtain the graphs of certain related functions.
Vertical and horizontal shifts:- suppose .0>c
( )y f x c= + , shift the graph of )(xfy = a distance of units upward.c
( )y f x c= − , shift the graph of )(xfy = a distance of units downward.c
)( cxfy −= , shift the graph of )(xfy = a distance of units to the right.c
)( cxfy += , shift the graph of )(xfy = a distance of units to the left.c
Vertical & Horizontal stretching and reflecting:- suppose 1>c
( )y c f x= , stretch the graph of )(xfy = vertically by a factor of .c
1 ( )y f xc= , compress the graph of )(xfy = vertically by a factor of .c
( )y f cx= , compress the graph of )(xfy = horizontally by a factor of .c
( )xy f c= , stretch the graph of )(xfy = horizontally by a factor of .c
)(xfy −= , reflect the graph of )(xfy = about the x- axis.
)( xfy −= , reflect the graph of )(xfy = about the y- axis.
Example 7:
Given the graph of xy = , use transformation to graph
xyxyxy −=−=−= ,2,2
xyxy −== ,2
Solution:-
9
10. Example 8:
Sketch the graph of the function 106)( 2
++= xxxf
Solution:-
Completing the square
1)3(106 22
++=++= xxxy
ii) Combination of functions:-
Algebra of functions
domain)()())(( xgxfxgf +=+ gf DD ∩
domain)()())(( xgxfxgf −=− gf DD ∩
domain)()())(( xgxfxfg = gf DD ∩
)(
)(
)(
xg
xf
x
g
f
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
domain 0)(, ≠xfDD gf ∩
Example 9:
if
2
4)(,) xxgxx −=(f = , find the functions
g
fandfggfgf ,,, −+ .
Solution:-
( )f x x= : is [0,0x ≥ ⇒ fD ∞ )
2
( ) 4g x x= − :
22 2
4 0 4 4x x x x− ≥ ⇒ ≤ ⇒ ≤ ⇒ ≤ 2
22 ≤≤− x ⇒ gD is [-2,2]
[ ]: 0,2f gD D∩
2
4f g x x+ = + − D: 20 ≤≤ x
2
4f g x x−− = − D: 20 ≤≤ x
2 2
4 4fg x x x x= − = − D: 20 ≤≤ x
2 4
x
x
f
g −
= D: 20 <≤ x
10
11. iii) Composition of function
Suppose that uufy == )( and 1)( 2
+== xxgu
( ) 1)1()()( 22
+=+=== xxfxgfufy
Def.: Given two functions andf g , the composite function f g (also called
the composition of andf g ) is defined by ( ) ( )( ) ( )f g x f g x=
The domain of f g is the set of all is in the domain ofx g such that is in
the domain of .
)(xg
f
Example 10:
2
)( xxf =If , and 3)( −= xxg , find the composite function f g and fg
Solution:-
( ) 2
( )( ) ( ) ( 3) ( 3)
f g
f g x f g x f x x D is= = − = − R
( ) 2 2
( )( ) ( ) ( ) 3
g f
g f x g f x g x x D is= = = − R
In general f g g f≠
Example 11:
If ( )f x x= and , ( ) 2g x x= − , find each function and its domain
a) f g b) fg c) f d)f g g
Solution:-
( )a) ( ) 4
( )( ) ( ) 2 2 2f g x f g x f x x x= = − = − = −
( ]: 2 0 2 , 2f g
D x x− ≥ ⇒ ≤ ⇒ −∞ ,
b) ( )( )( ) ( ) ( ) 2g f x g f x g x x= = = −
0x tobedefined x ≥ & 2 2 0x tobedefined x x 4− − ≥ ⇒ ≤
0 4x∴ ≤ ≤ : 0,4g fD ⎡ ⎤
⎣ ⎦
c)
1/ 2 1/ 2 1/ 4 4
( )( ) ( ( )) ( ) ( )f f x f f x f x x x x x= = = = = =
)0,f fD ⎡
⎣
∴ = ∞
d) ( )( ) ( ( )) ( 2 ) 2 2g g x g g x g x x= = − = − −
2 2 0tobedefinedx x− − ≥ 2x⇒ ≤
2 2 2 2 0 2 2 2 4x x x xtobedefined− − − − ≥ ⇒ − ≤ ⇒ − ≤ ⇒ ≥ −2x
[ ]2,2g gD −∴ =
11
-2
2
12. II TRIGONOMETRY
1. Trigonometric functions
Trigonometric function of any angle :x
c
b
a
x
sin cos tan
csc sec cot
, ,
, ,
a bx xc c b
c cx xa ab
= =
= =
ax
bx
=
=
Positive functions :
Values of basic Trigonometric functions:
3
2 2
3
2 2
sin(0) 0 , sin( ) 1 , sin( ) 0 , sin( ) 1 , sin(2 ) 0
cos(0) 1 , cos( ) 0 , cos( ) 1 , cos( ) 0 , cos(2 ) 1
π π
π π
π π
π π
= = = = −
= = = − =
=
=
Example 1:
Find the values of the trigonometric functions for 3
4
πθ =
Solution:-
3 3) )
4 4
3 3) 2 ) 2 )
4 4
1 1sin( cos( tan( 1
2 2
csc( sec( cot( 1
, ,
, ,
π π
π π
3 )
4
3
4
π
π
−= =
= − = = −
= −
12
13. 2. Graphs of the trigonometric functions
i) Y=sin(x):
(1) Odd Functions
(2) Domain is R
(3) Range [-1, 1]
(4) Period is 2π
ii) Y=cos(x):
(1) Even Functions
(2) Domain is R
(3) Range [-1, 1]
(4) Period is 2π
i) Y=tan(x):
(1) Odd Functions
(2) Domain is R-{±π /2, ±3π /2, ±5π /2........}
(3) Range R
(4) Period is π
y=csc x y=secx y=cotx
13
14. Example 2:
Sketch the graphs of the following functions
(i) Y = 2 sin (x) (ii) Y = Sin (2x)
Solution:-
i-
ii-
Note: In general if y = sin (n Ө) its Period is 2π /n
14
15. 3. Trigonometric identities
1- Sin2
x + Cos2
x =1
2- Tan2
x +1 = Sec2
x.
3- Cot2
x +1 = Cosec2
x
4- Sin (a ± b) = Sin (a) Cos (b) ± Cos (a) Sin (b)
5- Cos (a ± b) = Cos (a) Cos (b) ∓ Sin (a) Sin (b)
6- Sin (2a) = 2 Sin (a) Cos (a)
7- Sin2
(a) = (1/2)(1-Cos(2a))
8- Cos2
(a) = (1/2)(1+Cos(2a))
9- Cos (2a) = Cos2
(a)-Sin2
(a) = 1-2Sin2
(a) = 2Cos2
(a) -1
10- Tan (a ± b) = [(Tan (a) ± Tan (b)] /[1 Tan(a) Tan(b)]∓
4. Solving trigonometric equations
Example 3:
1
2sin( )θ = if
i = [0,2 ]θ π∈
ii = θ ∈R
Solution:-
[0,2 ]i- θ π∈
)6/5(,6/ πθπθ ==
ii- θ ∈ R
.....2,1,0,26/ ±±=+= nnππθ
,.....2,1,0,2)6/5( ±±=+= nnππθ
Example 4:
Solve 2
(1 3) ( 3 1) 0, [0,2 ]Sec Tanθ θ θ π− + + − = ∈
Solution:-
2
2
(1 ) (1 3) ( 3 1) 0
(1 3) 3 0
)( 3) 0
1, 3
/ 4, /3
Tan Tan
Tan Tan
Tan Tan
Tan Tan
θ θ
θ θ
θ θ
θ θ
θ π π
+ − + + − =
− + + =
− − =
= =
=
( 1
Example 5:
Solve Co ( ) (2 ) 0s x Cos x+ = [0,2 ]x π∈
Solution:-
=
2
2 ( ) ( ) 1 0
(2 ( ) 1)( ( ) 1) 0
( ) 1/ 2, 1
/3, ,5 /3
Cos x Cos x
Cos x Cos x
Cos x
x π π π
+ − =
− +
= −
=
15
16. Note 1:
[ ]
2 2
2 2 2
2 2 2
cos sin 1 (1)
1 tan sec (1) / cos
cot 1 csc (1)/sin
sin( ) sin cos cos sin (2)
sin( ) sin cos cos sin (3)
1
(3) (2): sin cos sin( ) sin( )
2
(2) : sin 2 2sin cos
cos( ) cos cos
x x
x x x
x x x
x y x y x y
x y x y x y
x y x y x y
In put y x x x x
x y x y
+ =
+ = ⇐
+ = ⇐
+ = +
− = −
+ = − + +
= =
+ =
[ ]
[ ]
2 2
2 2 2
sin sin (4)
cos( ) cos cos sin sin (5)
1
(5) (4): cos cos cos( ) cos( )
2
1
(5) (4): sin sin cos( ) cos( )
2
(4) : cos2 cos sin
(1) sin 1 cos cos2 2cos 1 co
x y
x y x y x y
x y x y x y
x y x y x y
In put y x x x x
From x x x x
−
− = +
+ = − + +
− = − − +
= = −
= − = − ⇒ 2
2 2 2 2
2
1
s (1 cos
2
1
(1) cos 1 sin cos2 1 2sin sin (1 cos2 )
2
tan tan tan tan
tan( ) (6) tan( )
1 tan tan 1 tan tan
2tan
(6) : tan 2
1 tan
2 )x x
From x x x x x x
x y x y
x y x y
x y x y
x
In put y x x
x
= +
= − = − ⇒ = −
+ −
+ = − =
− +
= =
−
:2Note
2 2 2
2
2
2
) cos( ) cos cos sin sin sin
sin( ) cos
tan( ) cot
cos( ) sin
I x x x
x x
x
x x
x x
π π π
π
π
π
+ = − = −
+
+ = = = −
+ −
11 1
2 2
1
2
1
2
2 , )
[ 2 ]
2
) sin sin ( )
sin 4 & 3 . (
2 sin 0,
sin
x
II x usecalculator to find
x is ve for x in th rd quad x
There isonly valuesof x suchthat x for x
But in general thereisinfinitenumber of xvaluesthat makes x
x
6
:
π
π θ π θ
π
π
θ −− +
−
−
− = +
= ⇒ = =
− =
= ∈
=
= − 11
6 6
7
6 6
23 35
6 611
6 13
6 6
19 31
6 67
6 5 17
6 6
2
0, 1, 2,....
2
2 2 ....
2 2 ....
2 2 ....
2 2 ....
n
where n or indetails
x n
x x
x
x x
x x
x
x x
π π
π π
π π
π
π π
π π
π
π π
π
π π
π π
π π
π π
π π
− −
− −
+
+
= ± ±
= = +
+ = + =
=
− = − =
+ = + =
=
− = − =
=
16
17. III GRAPHS OF SECOND-DEGREE EQUATION
1. Circles
Equation of a circle with centre (h, k) and radius r is
222
)()( rkyhx =−+−
2. Parabolas
Equation of a Parabola is 2
y ax bx c= + +
e.g:
If we interchange x and y in the equation , the result is2
axy = 2
ayx =
3. Ellipses
The curve with equation:
12
2
2
2
=+
b
y
a
x
is called an ellipse.
4. Hyperbolas
The curve with equation:
12
2
2
2
=−
b
y
a
x
is called a hyperbola.
17
