Lecture Two

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1

Section 2.1 – Functions and Graphs

Relations

A

relation

is any set of ordered pairs. The set of all first components of ordered pairs is called the domain of the relation and the set of second components is called the range of the relation.

Definition of a Function

A

function

is a relation between two variables such that to matches each element of a first set (called domain) to an element of a second set (called range) in such way that no element in the first set is assigned to two different elements in the second set.

The domain of the function is the set of all values of the independent variable for which the function is defined.

The range of the function is the set of all values taken on by the dependent variable.

Example

Determine whether each relation is a function and find the domain and the range.

a) F



(1, 2), ( 2, 4), (3, 1) 



Function: Yes Domain: {-2, 1, 3}

Range: {-1, 2, 4}

b) G



(1,1), (1, 2), (1,3), (2,3)



Function: No Domain: {1, 2}

Range: {1, 2, 3}

c) H 



( 4,1), ( 2,1), ( 2, 0) 



Function: No Domain: {-4, -2}

Range: {0, 1}

Domain

Range

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2 Example

Give the domain and range of each relation

Domain: {-1, 0, 1, 4}

Range: {-3, -1, 1, 2}

Domain:



^^{4, 4}



Range:



^^{6, 6}



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3 Functions as Equations y 

0.016

x2

0.93

x

8.5

x: independent y: depend on x

Notation for Functions

( )

f x _{read “}

f

of

x

” or “

f

at

x

” represents the value of the function at the number x.

Example

Let f x( ) x²5x3 a) f( )2

( )x x2 5 3 f    x

( ) ( )2 5( ) 3

f        ( )2 ( )2 2 5 )(2 3

f     ( )2 ^ 2 5* 2 3  

3 b) f( )q

( ) ( )q 2 5( ) 3 f q    q 

2 5 3

q q

   

Example

If f x

( )

x2 

2

x

7

, evaluate each of the following:

a) f( 5) b) f x( 4)

Solution

a) f(5)?

( ) ( )

2

2( ) 7

f      

(

5

) (

5

)

2

2(

5

) 7

f      

= 25 + 10 + 7 = 42

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4 b) f(x4)?

( ) ( )

2

2( ) 7

f      

(

4

) (

4

)

2

2(

4

) 7

f x  x  x  (ab)² a²2ab b ²

x²2(4)x4²2x 8 7

2 8 16 2 1

x x x

    

2 6 15

x x

  

Example

Let g x( )2x3, find g a( 1) Solution

( ) 2 3

g x  x

(a 1) 2(a 1) 3

g    

2a 2 3

   2a 5

 

Example

Given: f x( ) 2 x2 x 3, find the following.

a) f(0) b) f( 7) c) f(5 )a

Solution

a) f(x0)2( )0 2(0) 3 b) f(7)2(7)2(7) 3

= 108

c) f(5a)2(5a)2(5a) 3 50a2 5a 3

  

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5

Increasing and Decreasing Functions

A function rises from left to right (x-coordinate), the function f is said to be increasing on an open interval I (a, b) (x-coordinate)

( ) ( ) ab  f a  f b A function f is said to be decreasing on an open interval I

( ) ( ) ab  f a  f b A function f is said to be constant on an open interval I

( ) ( ) ab  f a  f b

Example

Determine the intervals over which the function is increasing, decreasing, or constant

Increasing: [1, 3]

Decreasing:



^^,1



Constant:



^3,^



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6 Example

State the intervals on which the given function f x

( )

 x3

3

x is increasing, decreasing, or constant

Increasing , 3 , 3,

2 2

  

   

   

    Decreasing ³, ³

2 2

 

 

 

Relative Maxima (um) and Minima (um)

f(a) is a relative maximum if there exists an open interval I about a such that f(a) > f(x), for all x in I.

f(a) is a relative minimum if there exists an open interval I about a such that f(a) < f(x), for all x in I.

The relative minimum value of the function is 1 @ x = /2 The relative maximum value of the function is 1 @ x = /2

3

 2

      



















x y

3 2

1

1

/2

/2

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7

Determine any relative maximum or minimum of the function, determine the intervals on which the function increasing or decreasing, and then find the domain and the range.

Relative Maximum:

Relative Minimum:

Increasing:

Decreasing:

Domain:

Range:

Relative Maximum:

Relative Minimum:

Increasing:

Decreasing:

Domain:

Range:

Relative Maximum:

Relative Minimum:

Increasing:

Decreasing:

Domain:

Range:

Relative Maximum:

Relative Minimum:

Increasing:

Decreasing:

Domain:

Range:

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8

Piecewise-Defined Functions

Function are sometimes described by more than one expression, we call such functions piecewise- defined functions.

Example

Graph each function

2 5 2

( ) 1 2

x if x

f x x if x

  

   

Find: f(2) 2(2) 5 1 (0) 2( )0 5 5

f    

4

(4) 1 5

f   

2 3 1

( ) 6 1

x if x

f x x if x

 

   

Example

20 if 0 60 ( ) 20 0.40( 60) if 60

C t

t

t t

  

    

Find C(40), C(80), and C(60) Solution

a) C(40) = 20

b) C(80) = 20 + 0.40(80 – 60) = 28 c) C(60) = 20

         





















x y

         























x y

(9)

9 Example

The speed s of sound in air is a function of the temperature t, in degrees Fahrenheit, and is given by 5 2457

( ) 1087.7

2457

s t  t

Where s is in feet per second. Find the speed of sound in air when the temperature is 0, 32.

Solution

s

(0) = 1087.7 ⁵⁽⁰⁾ ²⁴⁵⁷ 2457

 1087.7 / secft



5(32) 2457 (32) 1087.7

s  2457

1122.6 / secft



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10

Even and Odd Functions

Given the function f x( ) then find f(x) and simplify:

 If f(x) f( )x  f is even, or

 If f(x)



f x( ) f is odd, or

 Neither

Example

Decide whether each function is even, odd, or neither a) f x

( )



8

x4

3

x2

4 2

(

x

) 8(

x

) 3(

x

)

f     

4 2

8x 3x

 

( )

 f x Function is Even

b) 3

( ) 6 9

f x  x  x

(

x

) 6(

x

)

3

9( )

f     x

6x3 9x

  



⁶^x³ ⁹^x







 ( )

 f x

Function is Odd c) f x

( )



3

x2

5

x

(

x

) 3(

x

)

2

5( )

f     x

3x2 5x





Function is Neither

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11

Exercise Section 2.1 – Functions and Graphs

1. Determine whether each relation is a function and find the domain and the range.

a) {(1, 2), (3, 4), (5, 6), (5, 8)}

b) {(1, 2), (3, 4), (6, 5), (8, 5)}

c) {(9, -5), (9, 5), (2, 40)}

d) {(-2, 5), (5, 7), (0, 1), (4, -2)}

e) {( -5,3), (0, 3), (6, 3)}

2. Identify the domain and the range: {(5, 12.8), (10, 16.2), (15, 18.9), (20, 20.7), (25, 21.81)}

3. Let f x( )  3x 4, find f(0) 4. Let g x( ) x²4x1, find g(x) 5. Let f x( )  3x 4, find f a( 4)

6. Given: f(x)2|x|3x, find f(2h). 7. Given: ( ) ⁴

3 g x x

x

  , find g x h(  ) 8. Given:

( ) 2 1 g x x

x



 , find g(0) and g ( 1) 9. Given that ^{g x}

 

^²^x²^²^x^³^{. Find}^{g p}



^³



10. If f x

( )

 x2 

2

x

7

, evaluate each of the following: f( 5), ( f x4), (f x) 11. Find

     

^{0 ,} ^{4 ,} ^{7 ,}

 

³₂

^{ }

₂

16

g g g and g for g x x

x

 

 12.

2 4

( ) 4 2

3 2

x if x

f x x if x

x if x

  



    

 



Find: f( 5), ( 1), (0),  f  f and f (3)

13.

2 3

( ) 3 1 3 2

4 2

x if x

f x x if x

x if x

  



    

  



Find: f( 5), ( 1), (0),  f  f and f (3)

14.

3

2

3 2 0

( ) 3 0 1

4 1 3

x if x

f x x if x

x x if x

    

   

    



Find: f( 5), ( 1), (0),  f  f and f (3)

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12 15.

2 9 3

( ) 3

6 3

x if x

h x x

if x

 

 

 

 



Find: h(5), (0), h and h (3)

16. Graph the piecewise function defined by











 

1 2

1 ) 3

( x if x

x x if

f

17. Sketch the graph ³

2 1

( ) 1 1

3 1

x if x

f x x if x

x if x

  



   

  



18. Sketch the graph ²

3 2

( ) 2 1

4 1

x if x

f x x if x

x if x

  



    

  



19. Decide whether the function is even, odd, or neither f x( )  x³ 2x 20. Decide whether the function is even, odd, or neither f x( )x⁵2x³ 21. Decide whether the function is even, odd, or neither f x( ).5x⁴2x²6 22. Decide whether the function is even, odd, or neither f x( ).75x² x 4 23. Decide whether the function is even, odd, or neither f x( )x³ x 9 24. Decide whether the function is even, odd, or neither f x( )x⁴5x8 25. Decide whether each function is even, odd, or neither f x( )x³x 26. Decide whether each function is even, odd, or neither g x( )x²x 27. Decide whether each function is even, odd, or neither h x( )2x²x⁴

28. Decide whether each function is even, odd, or neither f x( )2x²x⁴1 29. Decide whether each function is even, odd, or neither ¹ ⁶ ²

( ) 5 3

f x  x  x

30. Decide whether each function is even, odd, or neither f x( )x 1x² 31. Decide whether each function is even, odd, or neither f x( )x² 1x² 32. Decide whether each function is even, odd, or neither f x

( )



5

x7

6

x3

2

x 33. Decide whether each function is even, odd, or neither

6 2

( ) 5 3 7

f x  x  x 

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13

34. Decide whether the function is even, odd, or neither f x( )x²6 35. Decide whether the function is even, odd, or neither f x( )7x³x 36. Decide whether the function is even, odd, or neither h x( )x⁵1

37. The elevation H, in meters, above sea level at which the boiling point of water is in t degrees Celsius is given by the function

( ) 1000(100 ) 580(100 )2

H t   t t

At what elevation is the boiling point 99.5.

38. A hot-air balloon rises straight up from the ground at a rate of 120 ft./min. The balloon is tracked from a rangefinder on the ground at point P, which is 400 ft. from the release point Q of the balloon.

Let d = the distance from the balloon to the rangefinder and

t

– the time, in minutes, since the balloon was released. Express d as a function of

t

.

39. An airplane is flying at an altitude of 3700 ft. The slanted distance directly to the airport is

d

feet.

Express the horizontal distance

h

as a function of

d

.

40. A device used in golf to estimate the distance d, in yards, to a hole measures the size s, in inches, that the 7-ft pin appears to be in a viewfinder. Express the distance d as a function of s.

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14

41. A rancher has 360 yd. of fencing with which to enclose two adjacent rectangular corrals, one for sheep and one for cattle. A river forms one side of the corrals. Suppose the width of each corral is x yards.

a) Express the total area of the two corrals as a function of x. b) Find the domain of the function.

42. A rhombus is inscribed in a rectangle that is w meters wide with a perimeter of 40 m. Each vertex of the rhombus is a midpoint of a side of the rectangle. Express the area of the rhombus as a function of the rectangle’s width.

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1

Solution Section 2.1 – Functions and Graphs

Exercise

Determine whether each relation is a function and find the domain and the range.

a) {(1, 2), (3, 4), (5, 6), (5, 8)}

b) {(1, 2), (3, 4), (6, 5), (8, 5)}

c) {(9, -5), (9, 5), (2, 40)}

d) {(-2, 5), (5, 7), (0, 1), (4, -2)}

e) {( -5,3), (0, 3), (6, 3)}

Solution

a) {(1, 2), (3, 4), (5, 6), (5, 8)}

Not a function Domain: {1, 3, 5}

Range: {2, 4, 6, 8}

b) {(1, 2), (3, 4), (6, 5), (8, 5)}

Function

Domain: {1, 3, 6, 8}

Range: {2, 4, 5}

c) {(9, -5), (9, 5), (2, 40)}

It is not a function Domain = {2, 9}

Range = {-5, 5, 40}

d) {(-2, 5), (5, 7), (0, 1), (4, -2)}

It is a function

Domain = {-2, 0, 4, 5}

Range = {-2, 1, 5, 7}

e) {( -5,3), (0, 3), (6, 3)}

It is a function Domain = {-5, 0, 6}

Range = {3}

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2

Exercise

Find the domain and the range of the relation: {(5, 12.8), (10, 16.2), (15, 18.9), (20, 20.7), (25, 21.81)}

Solution

Domain: {5, 10, 15, 20, 25}

Range: {12.8, 16.2, 18.9, 20.7, 21.81}

Exercise

Let f x( )  3x 4, find f(0) Solution

(0) 3(0) 4

f   

4

Exercise

Let g x( ) x²4x1, find g(x) Solution

( ) ( )2 4( ) 1

g x   x  x 

2 4 1

x x

   

Exercise

Let f x( )  3x 4, find f a( 4) Solution

(a 4) 3(a 4) 4

f     

3a 12 4

    3a 8

  

Exercise

Given: f(x)2|x|3x, find f(2h). Solution

(2 ) 2 | 2 | 3(2 ) f h  h  h

2 | 2 h| 6 3h

   

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3

Exercise

Given: ( ) ⁴

3 g x x

x

  , find g x h(  ) Solution

( ) 4

3 x h x h

g x h   

  

Exercise

Given:

1 2 ) (

x x x

g



 , find g(0) and g ( 1)

Solution

a) 2

0 1 0

(0) 0

g



 

b) 2

1 1

1 ( 1) 0

( 1)

g ^ ^ undefined

 

  

Exercise

Given that ^{g x}

 

^²^x²^²^x^³^{. Find}^{g p}



^³



Solution



³



²



^p ³



² ²



^p ³



³

g p     



^a^^b



² ^^a²^²^ab^^b²

  



² ²



2 p 2 p 3 3 2p 6 3

     



²



2 p 6p 9 2p 9

    

2p2 12p 18 2p 9

    

2p2 14p 27

  

Exercise

If f x

( )

x2 

2

x

7

, evaluate each of the following: f( 5), ( f x4), (f x) Solution

a. f(5)?

( )

2

2 7

f x  x  x

(18)

4

(

5

) (

5

)

2

2(

5

) 7

f      

= 25 + 10 + 7 = 42

b. f(x4)?

(

4

) (

4

)

2

2(

4

) 7

f x  x  x  (a b )² a²2ab b ²

x²2(4)x4²2x 8 7

2 8 16 2 1

x x x

    

2 6 15

x x

  

c. ^f

⁽

^^x

⁾

^

 

^^x ²^

²  

^^x ^

⁷

2 2 7

x x

  

Exercise

Find

     

0 , 4 , 7 ,

 

³2

^{ }

2 16

g g g and g for g x x

x

 

 Solution

 

2

0 0

0 6

6

0 0 1 1

g   



   

² ^'

4 4

1 0 4 4

4 6 16

16

doesn t e

g      xi ts

 



 

 

2

7 7

16 49 33

7

7 16

7 doesn t exist in real numb'

g     er

 



 

³²

 

³ ²

3 2 16 2

g 

 3 2

9 16 4





 

4 16 9 4 3 2

 

55 4 3

 2

(19)

5 55

2 3

 2

3 2

2 55

  3 55



3 55

 55

Exercise

2 4

( ) 4 2

3 2

x if x

f x x if x

x if x

  



    

 



Find: f( 5), ( 1), (0),  f  f and f (3)

Solution

a) f( 5) 25 3 b) f( 1)  ( 1)1 c) f(0)00 d) f(3)3(3)9

Exercise

2 3

( ) 3 1 3 2

4 2

x if x

f x x if x

x if x

  



    

  



Find: f( 5), ( 1), (0),  f  f and f (3)

Solution

a) f(5) 2(5)10 b) f(1)3(1)1 4 c) f( )0 3( )0 1 1 d) f( )3  4( )3  12

(20)

6

Exercise

3

2

3 2 0

( ) 3 0 1

4 1 3

x if x

f x x if x

x x if x

    

   

    



Find: f( 5), ( 1), (0),  f  f and f (3)

Solution

a) f(5) doesn t exist' b) f(1)(1)³32 c) f( )0 ( )0 ³33 d) f( )3  4 ( )3 ( )3 ² 2

Exercise

2 9 3

( ) 3

6 3

x if x

h x x

if x

 

 

 

 



Find: h(5), (0), h and h (3) Solution

a) ( )5 ⁵² 9 3

5 8

h  

  b) (0) ⁰² 3

3 0

h 9

 

 c) h(3)6

Exercise

Graph the piecewise function defined by











 

1 2

1 ) 3

( x if x

x x if

f Solution

      



















x y

(21)

7

Exercise

Sketch the graph ³

2 1

( ) 1 1

3 1

x if x

f x x if x

x if x

  



   

  



Solution

Exercise

Sketch the graph ²

3 2

( ) 2 1

4 1

x if x

f x x if x

x if x

  



    

  



Solution

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8

Exercise

Decide whether each function is even, odd, or neither f x( )  x³ 2x Solution

( x) ( x)3 2( ) f      x

3 2

x x

  ( )

 f x  The function is odd.

Exercise

Decide whether each function is even, odd, or neither f x( )x⁵2x³ Solution

5 3

( ) ( x) 2( ) f x    x

5 2 3

x x

   ( )

Exercise

Decide whether each function is even, odd, or neither f x( ).5x⁴2x²6 Solution

4 2

( ) .5( ) 2( ) 6

f x  x  x 

4 2

.5x 2x 6

  

( )

 f x  The function is even.

Exercise

Decide whether each function is even, odd, or neither f x( ).75x² x 4 Solution

( x) .75( )2 x 4 f   x   

.75x2 x 4

  

( )

(23)

9

Exercise

Decide whether each function is even, odd, or neither f x( )x³ x 9 Solution

( x) ( x)3 ( x) 9 f      

3 9

x x

     The function is neither.

Exercise

Decide whether each function is even, odd, or neither f x( )x⁴5x8 Solution

( ) ( x)4 5( ) 8 f x    x 

4 5 8

x x

    The function is neither.

Exercise

Decide whether each function is even, odd, or neither f x( )x³x Solution

( x) ( x)3 ( ) f     x

x3 x

   ( )

Exercise

Decide whether each function is even, odd, or neither g x( )x²x Solution

( x) ( x)2 ( ) g     x

x2 x

   The function is neither.

Exercise

Decide whether each function is even, odd, or neither h x( )2x²x⁴ Solution

2 4

( x) 2( x) ( ) h     x

(24)

10

2 4

2x x

 

( )

h x  The function is even.

Exercise

Decide whether each function is even, odd, or neither f x( )2x²x⁴1 Solution

2 4

( ) 2( x) ( ) 1

f x    x 

2 4

2x x 1

  

( )

Exercise

Decide whether each function is even, odd, or neither ¹ ⁶ ²

( ) 5 3

f x  x  x Solution

6 2

1

( x) 5( x) 3( x)

f     

6 2

1

5x 3x

 

( )

Exercise

Decide whether each function is even, odd, or neither f x( )x 1x² Solution

( x) x 1 ( )2 f    x

1 2

x x

   ( )

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11

Exercise

Decide whether each function is even, odd, or neither f x( )x² 1x² Solution

2 2

( ) ( x) 1 ( x) f x    

2 2

1

x x

 

( )

Exercise

Decide whether each function is even, odd, or neither f x

( )



5

x7

6

x3

2

x Solution

7 3

(

x

) 5(

x

) 6( ) 2( )

f     x  x

7 3

5x 6x 2x

   

7 3

(5

x

6

x

2 )

x

   

( )

  f x

 The function is odd.

Exercise

Decide whether each function is even, odd, or neither 6 2

( ) 5 3 7

f x  x  x  Solution

6 2

( ) 5( ) 3( ) 7

f x  x  x 

6 2

5x 3x 7

  

( )

 f x

 The function is even.

Exercise

Decide whether each function is even, odd, or neither f x( )x26 Solution

( x) ( x)2 6

f    

2 6

x 

( )



f x

 The function is even.

(26)

12

Exercise

Decide whether each function is even, odd, or neither f x( )7x3x Solution

( ) 7( )3 ( ) f x  x  x

7x3 x

   (7x3 x)

  

( )



 f x

 The function is odd.

Exercise

Decide whether each function is even, odd, or neither h x( )x51 Solution

( x) ( x)5 1

h    

5 5

5

1 1

( 1)

x x

x

Neither

  

   

  



Exercise

The elevation H, in meters, above sea level at which the boiling point of water is in t degrees Celsius is given by the function

( ) 1000(100 ) 580(100 )2

H t   t t

At what elevation is the boiling point 99.5.

Solution

99.5 99 2

( ) 1000(100 .5) 580(100 99.5)

H    

645 m



(27)

13

Exercise

A hot-air balloon rises straight up from the ground at a rate of 120 ft./min. The balloon is tracked from a rangefinder on the ground at point P, which is 400 ft. from the release point Q of the balloon. Let d = the distance from the balloon to the rangefinder and

t

– the time, in minutes, since the balloon was released.

Express d as a function of

t

. Solution

2 2

2 (120t) 400

d

2 2 400 )

120 ( )

(t  t 

d

160000 14400 ²

 t

d

) 100 9

( 1600 ²

 t

d

( ) 40 9 2 100

d t  t 

Exercise

An airplane is flying at an altitude of 3700 ft. The slanted distance directly to the airport is

d

feet. Express the horizontal distance

h

as a function of

d

.

Solution

2 2 2 (3700) h

d  

2 2 2 (3700) h

d  

2 2 (3700) )

(t  d  h

Exercise

A device used in golf to estimate the distance d, in yards, to a hole measures the size s, in inches, that the 7-ft pin appears to be in a viewfinder. Express the distance d as a function of s.

Solution in

ft s

d 7

6  s in

d in

ft6

 7

ft ft yd d s

3 42 1

 ( ) 14 d s  s

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14

Exercise

A rancher has 360 yd. of fencing with which to enclose two adjacent rectangular corrals, one for sheep and one for cattle. A river forms one side of the

corrals. Suppose the width of each corral is x yards.

a) Express the total area of the two corrals as a function of x.

b) Find the domain of the function.

Solution

a) Express the total area of the two corrals as a function of x.

360 3  

 x l P

x l3603

) 3 360

( x

x xl

A  

360 2

( ) 3

A x x x

  

b) Find the domain of the function.

0 ) 3 360

(  x 

x

0

x 3603x03x360x120 Domain: 0 < x < 120

Exercise

A rhombus is inscribed in a rectangle that is w meters wide with a perimeter of 40 m. Each vertex of the rhombus is a midpoint of a side of the rectangle. Express the area of the rhombus as a function of the rectangle’s width.

Solution

The area of the rhombus = 1

2 area of the rectangle, since each vertex of the rhombus is a midpoint of a side of the rectangle.

: Perimter

2l2w40 Divide both sides by 2 20

l w 20 l w



²⁰



Area of the rectanglelw w w



²



1 2 20 Area of the rhombus ww

1 2

2w 10w

  