Math for 800 09 functions

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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

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