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CONTENTS
ALGEBRAIC FUNCTIONS
Functiona relationship or
expression involving one or more variables.
DEFINITION OF FUNCTION
A function f from a set
A to a set B is a
relation that assigns to
each element x in the
set A exactly one
element on the set B.
Set A
Set B
DOMAIN AND RANGE
The set A is the
domain of the function f, and the
set B contains the
range.
Set A Set B
DOMAIN OF A FUNCTION
real numbers
, 0h x
f x g xg x
2
, 2 5 02 5
xf x x
x
5
2Df
DOMAIN
, 0f x g x g x
5 1, 5 1 0f x x x
1,
5Df x x
DOMAIN
, 0
h xf x g x
g x
2 1
, 5 1 05 1
xf x x
x
1,
5Df x x
DOMAIN
REPRESENTING A FUNCTION
Some algebraic expressions are called
functions and are represented by f (x).
The symbol “f (x)” do not represent a
product; is merely the symbol for an
expression, and is read “f of x”.
REPRESENTING A FUNCTION
Verbally: by a sentence.
Numerically: by a table.
REPRESENTING A
FUNCTION
Algebraically: by an
equation in two variables.
REPRESENTING A
FUNCTION
Graphically: By points on
a graph in a coordinate
plane.
REPRESENTING A
FUNCTION
The symbol f (x)
corresponds to the y−value
for a given x, y = f(x).
ALGEBRAIC FUNCTIONS
GRAPH OF A FUNCTION
GRAPH OF A FUNCTION
The graph of a function f is the
collection of ordered pairs
(x, f(x)) such that x is in the
domain of f.
x : distance from y-axis.
f(x) : distance from x-axis. ,x f x
INTERCEPTS OF A FUNCTION
To find the x−intercept(s), let
y = f (x) = 0 and solve the
equation for x.
To find the y−intercept(s), let
x = 0 and solve the equation
for y. 0y f x
y
x
y
x
y
x
Symmetry to the y-axis Symmetry to the origin Symmetry to the x-axis
(Not a function)
(-x, y) (x, y)(x, y)
(-x, -y)
(x, y)
(x, -y)
SYMMETRY OF A FUNCTION
y
x
Maximum
a
f (a)
y
x
Relative
Maximum
a
f (a)
x1 x2
MAXIMUM OF A FUNCTION
y
x
Minimum
a
f (a)
y
x
Relative
Minimum
x1 x2
a
f (a)
MINIMUM OF A FUNCTION
TYPES OF FUNCTIONS
LINEAR FUNCTION
A linear function is defined by , where m and
b are real numbers.
m: slope of the line
b: y−intercept
f x mx b
y
x
b
rise
run
y mx b
21
3y x
2
3
1
x
y
slopey-intercept
14
2y x
2
-1
4
x
y
slopey-intercept
LINEAR INEQUALITIES
GRAPH A LINEAR INEQUALITY
1.Rearrange the
equation so " " is on the left and
everything else on
the right.
2 3 6x y
3 2 6
22
3
y x
y x
GRAPH A LINEAR INEQUALITY
2. Plot the "y=" line (a solid line for
y≤ or y≥, and a
dashed line for
y< or y>).
22
3y x
22
3y x
GRAPH A LINEAR INEQUALITY
3. Shade above the
line for a "greater
than" (y> or y≥) or
below the line for a
"less than" (y< or y≤).
22
3y x
22
3y x
22
3y x
22
3y x
LINEAR INEQUALITY
QUADRARTIC FUNCTION
A quadratic function is a
function described by an
equation that can be written in
the form:
2f x ax bx c 0a where
vertex(Xv, Yv)
x
y
VERTEX
The graph of any quadratic
function is a parabola.
24
2 4v v
b ac bX Y
a a
MINIMUM
If a > 0 the parabola
opens upwards and
the vertex is the lowest
point of the parabola
(minimum).
2f x ax bx c Vertex (minimum)
MAXIMUM
If a < 0 the parabola
opens downwards and
the vertex is the highest
point of the parabola
(maximum).
Vertex (maximum) 2f x ax bx c
f(x)=(x-1)^2+3
f(x)=-(x+1)^2-3
Series 1
-12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16
-6
-4
-2
2
4
6
8
x
y
vertex(maximum)
a < 0
vertex(minimum)
a > 0
2f x ax bx c
EXTREME VALUES
x
y
x1 x2
(Xv, Yv)
2
0
0
f x
ax bx c
2 4
2
b b acx
a
ROOTS
x
y
x1 x2
(Xv, Yv)
1 2
2
x xXv
VERTEX
EQUATION OF A CIRCLE
The equation
of a circle with center at (h, k)
and radius r is:
2 2 2x h y k r
EQUATION OF A CIRCLE
The equation
of a circle with center at (0, 0)
and radius r is:
2 2 2x y r
GRAPH OF A FUNCTION
SPECIAL FUNCTIONS
SPECIAL FUNCTIONS
Special symbols are used
to represent some
defined functions.
2 21 1f x x x x
, , # #
c ca b a b
Z a b c a b cb c b c
2 5f x x
2 5x x
2 5x x
2 5x x
If , find . 2 5x x 3 5
23 3 5 14
25 5 5 30
3 5 14 30 44
2, 5f x y x y y
2, 5x y x y y
2, 5x y x y y
2, 5x y x y y
If , find . 2, 5x y x y y 4,3
24,3 4 3 5 3
48 15 63
SPECIAL FUNCTIONS
SUMMARY
