2016 1 functions

Functions and Graphs Mathematics Section UniKL MFI Norhayati Bakri

Definition of a Function Function as a Machine Function defined by an equation Evaluation of Functions

Domain of Function Graphs of Functions Intercepts The Vertical Line Test Piecewise Functions

Definition of a Function A function is a rule that assigns to each element in set X an element in set Y set Y (rule) codomain set X

f

x

domain

f(x)

a f(a) b

output

c input

f(b)=f(c)

f : x ď‚Ž f ( x) f : input ď‚Ž output

range

Injection Moulding Machine

nylon

polythene

DVD / CD storage

Function as a Machine

3

Function can be displayed in tabular form x

Multiply by 4

Add 5

f(x)

3

17

4

21

5

25

Function can be displayed as ordered pairs { ( 3 , 17 ) , ( 4 , 21 ) , ( 5 , 25 ) }

17

A function must assign to each input a UNIQUE output

3 main components in a Function input

output

relationship

A function relates an input to an output.

A

A 2K 3K

B

C

C 4K 5K

D employee

2K 3K

B

Function

D

5K

employee

salary

A

4K salary

A

2K 3K

B

4K 5K

C employee

salary

2K

B C

NOT Function

4K

D employee

salary

RELATION between the employee and their salary in company XY

Function defined by an equation A function can be defined implicitly: 5x  2y  3  0 If solve for y:

2y  5x  3

Now it can be defined explicitly:

y

1 (5 x  3 ) 2

1 f ( x )  (5 x  3 ) 2

‘x’ is considered as the input variable (independent variable) ‘y=f(x)’ is considered as the output variable (dependent variable)

Worked example: Does the following equation define y as a function of x? x 2 y  16

Solve for y 16 y 2 x

xy 2  16

Solve for y 16 y  x

x2 y  0

When solved, it’s either

2

x can be any real number except 0

x must be any real number greater than 0

FUNCTION

NO

x  0 or y  0

NO

Evaluation of Function To replace or to substitute the function’s variable with a number or an expression.

Consider f (x)  1 x  x2

If x  2 If x  h  3

f (2)  1 (2)  (2)2  1 f (h  3)  1 (h  3)  (h  3)2  1 h  3  (h2  6h  9)  1 h  3  h2  6h  9  h2  5h  5

Domain of Function Domain of a function is the largest possible set of inputs into the function.

• Domain of polynomial functions is all the real numbers. • Positive radical functions cannot contain a negative number underneath the radical. Set the radicand greater than or equal to zero and solve for the variable. •

Rational functions cannot have zeros in the denominator.

Set the denominator to zero and solve for the variable. Then remove the variable from the domain.

Worked example: Domain of polynomial functions is all the real numbers.

What is the domain of f ( x)  4x  2 ?

18

14

10

f(x) is a linear function (polynomial) Df  ( ,  )

6

2 -3

-2

-1

-2

-6

0

1

2

3

4

5

6

Worked example: Rational functions cannot have zeros in the denominator.

Set the denominator to zero and solve for the variable. Then remove the variable from the domain. x 1 What is the domain of g( x )  ?

4x  2

2 1.5 1

g(x) is a rational function 4x  2  0 x   21

0.5 0 -4

-3

-2

-1

0

-0.5 -1

Dg  ( , 21 )  (  21 , )

-1.5

1

2

3

4

Worked example: Positive radical functions cannot contain a negative number underneath the radical. Set the radicand greater than or equal to zero and solve for the variable.

5

What is the domain of h( x)  4x  2 ?

4

3

h(x) is a positive radical function 4x  2  0

2

x   21

1

0

Dh  [ 21 , )

-1

0

1

2

3

4

5

Worked example: What is the domain of J( x ) 

x 1 ?

4x  2 1.4

4x  2  0 x   21

1.2

1

0.8

DJ  ( 21 , )

0.6

0.4

0.2

0 -1

-0.5

0

0.5

1

1.5

2

Worked example: x 1 What is the domain of K( x )  ?

4x  2

Numerator:

x 1 0

x  1

[1, )

Denominator: 4x  2  0 x   21 DJ  [1, 21 )  ( 21 , )

Graphs of Functions For a function defined as y  f ( x) , we may view •

the inputs x as elements of a horizontal line in the plane

•

the outputs y as elements of a vertical line in the plane The graph of a function in the Cartesian plane is the set of values (x,f(x))

Graphs of functions y=f(x) can be sketched by creating a table of values (x,y) and plotting points on the Cartesian plane.

Worked example: a(x)

Graph a( x )  x

2.5

2

x

a(x)=| x |

-2

2

-1

1

-0.5

0.5

0

0

1

1

2

2

1.5

1

0.5

0 -3

-2

-1

0

1

2

Graph of absolute value function

3

Worked example: b(x) 3

Graph b( x)  x 3

2

x

b(x)

-8

-2

-1

-1

0

0

1

1

8

2

1

0 -9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

-1

-2

-3

Graph of cube root function

6

7

8

9

Characteristics of some simple functions Constant Function

f(x)

f(x)

f ( x)  c x

Linear Function

f(x)

x f(x)

m<0

f ( x)  mx  c

m>0

x

Quadratic Function

f ( x)  ax 2  bx  c

f(x)

x

f(x)

a<0 x

These characteristics help you to graph the function faster …

a>0 x

Intercepts • •

the x-intercept is a point where the graph crosses the x-axis the y-intercept is a point where the graph crosses the y-axis c(x) 6

x-intercept (3,0)

5

From the graph of

4

3 2

c( x)  x  2x  3 2

1 0 -3

x-intercept (-1,0)

-2

-1

-1 -2 -3 -4 -5

y-intercept (0,-3)

0

1

2

3

4

5

Worked example: Determine the intercepts of f ( x)  x 4  16 x2 To determine the y-intercept, compute f(0)

f(x) 250

f (0)  (0)  16(0)  0 4

2

The y-intercept is (0,0)

200

To determine the x-intercept, solve f(x)=0

150

100

x 4  16 x 2  0 50

x ( x  4)(x  4)  0 2

The x-intercepts are (-4,0) , (0,0) , (4,0)

0 -6

-5

-4

-3

-2

-1

0 -50

1

2

3

4

5

6

The Vertical Line Test In the definition of a function, it is required for a function to have a UNIQUE output for each input. â&#x20AC;˘

If the vertical line touches the graph at exactly one point at the y value of f(x), then the graph is a graph of a function.

â&#x20AC;˘

If the vertical line touches the graph at more than one point at the y value of f(x), then the graph is NOT a graph of a function.

Worked example: Determine which of the following graph is the graph of a function. g(x)

f(x)

intersection

4.00

40

3.00

30

2.00

20

1.00

10

0.00 0

1

2

3

4

5

6

7

8

9

-1.00

10

intersection

0 -4

-3

-2

-1

0

2

-10

-2.00

-20

-3.00

-30

-4.00

-40

Graph of NOT a function

1

Graph of a function

3

4

Piecewise Functions

A piecewise function is a function that obey one rule in one region and another rule in another region. Also known as split function.

5

Rules

Regions

4 3

 2x  1 , x  0 f ( x)   2 , 0x2 x

2 1 0 -2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-1 -2 -3 -4

A two-piece piecewise function

Worked example: Consider the following piecewise function and determine f(0), f(1) and f(3).  2x  2 , x  1 f ( x)    2  x , 1 x  3

When x<1, the function obeys f ( x)  2x  2 f (0)  2(0)  2  2

When 1≤x<3, the function obeys f ( x)  2  x f (1)  2  (1)  1 f (3)  undefined

Worked example: Draw the following piecewise function Rules

x  2  f (x)    1  1 x2 2

Regions

,  4  x  2 , 2 x 0 , x0 , 0x3

f ( x)  1 f ( x )  x

x  2  f (x)    1  1 x2 2

f ( x)  2

f ( x )  21 x 2

,  4  x  2 , 2 x 0 , x0 , 0x3

 4  x  2

2 x 0 x0

0x3

Swim with the current Be a good navigator Stay calm under pressure

Thank You See you next week