- Functions - Domain and Range - Linear Function - Quadratic Function - Special Functions

2. edwinxav@hotmail.com
elapuerta@hotmail.com

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

59. edwinxav@hotmail.com
elapuerta@hotmail.com