The document defines and explains various types of algebraic functions including linear, quadratic, and special functions. It provides definitions for key function concepts like domain, range, intercepts, maxima, minima, and symmetry. Examples are given for representing functions algebraically, numerically, graphically, and verbally. Specific functions like linear, quadratic, and special functions like absolute value are defined by their algebraic expressions and graphical properties like shape of their graphs. Methods for solving problems involving various function types are also demonstrated through examples.

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3. CONTENTS

4. ALGEBRAIC FUNCTIONS

5. Function
a relationship or expression involving one or more variables.

6. DEFINITION OF FUNCTIONA function ffrom a set Ato a set Bis a relation that assigns to each element xin the set Aexactly one elementon the set B. Set ASet B

7. DOMAIN AND RANGE
The set Ais the domainof the function f, and the set Bcontains the range.
Set A
Set B

8. DOMAIN OF A FUNCTIONreal numbers

9.  
 
 
,   0
h x
f x g x
g x
 
 
2
, 2 5 0
2 5
x
f x x
x

  

5
2
Df
 
    
 
DOMAIN

10. f  x  g  x, g  x  0
f  x  5x 1, 5x 1 0
1
,
5
Df x x
 
    
 
DOMAIN

11.  
 
 
,   0
h x
f x g x
g x
 
 
2 1
, 5 1 0
5 1
x
f x x
x

  

1
,
5
Df x x
 
    
 
DOMAIN

12. 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 “fof x”.

13. REPRESENTING A FUNCTION
Verbally: by a sentence.

14. Numerically: by a table. REPRESENTING A FUNCTION

15. Algebraically: by an equation in two variables. REPRESENTING A FUNCTION

16. Graphically: By points on a graph in a coordinate plane. REPRESENTING A FUNCTIONThe symbol f (x) corresponds to the y−value for a given x, y = f(x).

17. ALGEBRAIC FUNCTIONS

18. GRAPH OF A FUNCTION

19. 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

20. 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. y  f x  0

21. 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

25. y
x
Maximum
a
f (a)
y
x
Relative
Maximum
a
f (a)
x1 x2
MAXIMUM OF A FUNCTION

26. y
x
Minimum
a
f (a)
y
x
Relative
Minimum
x1 x2
a
f (a)
MINIMUM OF A FUNCTION

28. TYPES OF FUNCTIONS

31. 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

32. 2
1
3
y  x 
2
3
1
x
y
slope
y-intercept

33. 1
4
2
y   x 
2
-1
4
x
y
slope
y-intercept

34. LINEAR INEQUALITIES

35. GRAPH A LINEAR
INEQUALITY
1.Rearrange the
equation so " " is
on the left and
everything else on
the right.
2x 3y  6
3 2 6
2
2
3
y x
y x
  
  

36. GRAPH A LINEAR
INEQUALITY
2. Plot the "y=" line
(a solid line for
y≤ or y≥, and a
dashed line for
y< or y>).
2
2
3
y   x 
2
2
3
y   x 

37. 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≤).
2
2
3
y   x 

38. 2
2
3
y   x 
2
2
3
y   x 
2
2
3
y   x 
LINEAR INEQUALITY

41. QUADRARTIC FUNCTION
A quadratic function is a
function described by an
equation that can be written in
the form:
  2 f x  ax bx c where a  0

42. vertex
(Xv, Yv)
x
y
VERTEX
The graph of any quadratic
function is a parabola.
2 4
2 4 v v
b ac b
X Y
a a

  

43. MINIMUM
If a > 0 the parabola
opens upwards and
the vertex is the lowest
point of the parabola
(minimum).
f x  ax2 bx c Vertex (minimum)

44. MAXIMUM
If a < 0 the parabola
opens downwards and
the vertex is the highest
point of the parabola
(maximum).
  Vertex (maximum) 2 f x  ax bx c

45. 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
f x  ax2 bx c
EXTREME VALUES

46. x
y
x1 x2
(Xv, Yv)
 
2
0
0
f x
ax bx c

  
2 4
2
b b ac
x
a
  
 ROOTS

47. x
y
x1 x2
(Xv, Yv)
1 2
2
x x
Xv


VERTEX

48. EQUATION OF A CIRCLE
The equation
of a circle with
center at (h, k)
and radius r is:
    2 2 2 x h  y k  r

49. EQUATION OF A CIRCLE
The equation
of a circle with
center at (0, 0)
and radius r is:
2 2 2 x  y  r

50. GRAPH OF A FUNCTION

51. SPECIAL FUNCTIONS

52. SPECIAL FUNCTIONS
Special symbols are used
to represent some
defined functions.
 2 2 f x  x 1 x  x 1
 
   
, , # #
c c
a b a b
Z a b c a b c
b c b c
 
 
 

53.   2 f x  x 5
2 x  x 5
2 x  x 5
2 x  x 5

54. If 2 , find . x  x 5 3  5
2 3  3 5 14
2 5  5 5  30
3  5 14 30  44

55.   2 f x, y  x y 5y
2 x, y  x y 5y
2 x, y  x y 5y
2 x, y  x y 5y

56. If , find . 2 x, y  x y 5y 4,3
2 4,3 4 3 5 3
48 15 63
   
  

57. SPECIAL FUNCTIONS

58. SUMMARY

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