FACULTY OF SCIENCE, TECHNOLOGY AND HUMAN DEVELOPMENT

UNIVERSITI TUN HUSSEIN ONN MALAYSIA

BWM10103

TUTORIAL 1

Function, Domain and Range

1. Use the terms domain, range, independent variable, and dependent variable to explain how a function relates one variable to another variable.

2. Explain how the vertical line test is used to detect functions. 3. Does the independent variable of a function belong to the domain or range? Does the dependent variable belong to the

domain or range?

4. Determine the domains of the functions:

(a) 1

2y

x

(b)

2

1

9y

x

(c)

2 4

xy

x

5. Given 2

1( ) ,

2

xf x

x

find (a) (0);f (b) ( 1)f ; (c) (2 )f a ; (d) (1/ )f x

6. Sketch the graph of the function

5, 0 1

10, 1 2( )

15, 2 3

20, 3 4

x

xf x

x

x

.

Then, determine the domain and range of the function.

7. Let 2( ) 2 3f x x x . Evaluate (a) (3)f ; (b) ( 3)f ; (c) ( )f x (d) ( 2)f x (e) ( 2)f x .

8. Draw the graphs of the following functions, and find their domains and ranges.

(a) 2( ) 1f x x (b)

1, 0 1( )

2 , 1

x xf x

x x

(c)

2 4( )

2

xf x

x

(d)

2( ) 5f x x

9. Determine the domain of each of the following functions:

(a)2 4y x (b) 2 4y x (c) 2 4y x (d)

3

xy

x

(e)

2

( 2)( 1)

xy

x x

(f) 2

1

9y

x

(g)

2

2

1

1

xy

x

(h)

2

xy

x

Answer:

4. (a)The function is defined for every value of x except 2 i.e 2x

(b) The function is defined for 3x (c) Since 2 4 0x for all x , the domain is the set of real numbers.

5. (a) 1

2 (b)

2

3 (c)

2

2 1

4 2

a

a

(d)

2

21 2

x x

x

6. Domain: set of all positive real numbers; Range: set of integers, 5,10,15,20

7. (a) 6 (b) 18 (c) 2 2 3x x (d) 2 2 3x x (e) 2 6 11x x

8. (a) Domain: all numbers, Range: 1y (b) Domain: 0x , Range: 1 y or 2y

(c) Domain: 2x , Range: 4y (d) Domain: all numbers, Range: 5y

9. (a), (b), (g) all values of x ; (c) 2x ; (d) 3x ; (e) 1,2x ; (f) 3 3x ; (h) 0 2x

Limits

1. If ( ) 2 5f x x , examine what happens to the function as the value of x get closer and closer to 3.

2. Find the one-sided limits if 3, 1

( )2, 1

xf x

x

3. Find the one-sided limits if 1, 1

( )0, 1

xf x

x

4. Find the one-sided limits at x a if they exist. (a). 1a (b) 2a

5. Find the one-sided and two-sided limits at x a if they exist. (a). 1a (a). 1a

6. Find

(a) 2

lim5x

x

(b) 2

lim (2 3)x

x

(c) 22

lim ( 4 1)x

x x

(d) 3

2lim

2x

x

x

(e)

2

22

4lim

4x

x

x

(f)

2

4lim 25x

x

(g) 2

5

25lim

5x

x

x

(h)

24

4lim

12x

x

x x

(i)

3

23

27lim

9x

x

x

(j)

2

22

4lim

3 5x

x

x

(k)

2

21

2lim

1x

x x

x

7. Find

(a) 3 2

lim9 7x

x

x

(b)

2

2

6 2 1lim

5 3 4x

x x

x x

(c)

2

3

2lim

4 1x

x x

x

(d)

3

2

2lim

1x

x

x

(e)

3

2

2lim

1x

x

x (f) 5 4lim 7 2 5

xx x x

(g) 5 4lim 7 2 5

xx x x

8. Evaluate the following limits:

(a) 2

2lim( 4 )x

x x

(b) 3 21

lim( 2 3 4)x

x x x

(c)

2

1

3 1lim

1x

x

x

(d)

2

21

3 2lim

4 3x

x x

x x

(e) 22

2lim

4x

x

x

(f)

22

2lim

4x

x

x

9. Let ( ) 1f x x if 4x and 2( ) 4 1f x x x if 4x . Find

(a) 4

lim ( )x

f x

(b) 4

lim ( )x

f x

(c) 4

lim ( )x

f x

-1

2

-2

1

2

4

1 -1

1

2

10. Let ( ) 10 7f x x if 1x and ( ) 3 2f x x if 1x . Find

(a) 1

lim ( )x

f x

(b) 1

lim ( )x

f x

(c) 1

lim ( )x

f x

Answer:

1. The limit of the function equals 11 as x approaches 3.

2. 1

lim ( ) 2x

f x

and 1

lim ( ) 3x

f x

3. 1

lim ( ) 0x

f x

and 1

lim ( ) 1x

f x

4. (a)1

lim ( ) 0x

f x

and 1

lim ( ) 2x

f x

4. (b) 2

lim ( ) 1x

f x

and 2

lim ( ) 1x

f x

5. (a)1

lim ( ) 2x

f x

and 1

lim ( ) 4x

f x

1

lim ( )x

f x

does not exist

(a)1

lim ( ) 1x

f x

and 1

lim ( ) 2x

f x

1

lim ( )x

f x

does not exist

6. (a) 10; (b) 7; (c) -3; (d) 1

5 ; (e) 0; (f) 3; (g) -10; (h)

1

7; (i)

9

2; (j) 6; (k) The limit does not exist

7. (a) 1

3; (b)

6

5; (c) 0; (d) ; (e) ; (f) ; (g)

8. (a) -4; (b) 0; (c) 1

2; (d)

1

2; (e) 0; (f) , no limit

9. (a) 1; (b) 1; (c) 1

10. (a) 3; (b) 5; (c) The limit does not exist.

Continuity

1. Determine whether the following functions are continuous at a .

(a)

2

2

2 3 1( )

5

x xf x

x x

; 5a (b) ( ) 2; 1f x x a

(c)

2 1, 1

( ) ; 11

3, 1

xx

f x ax

x

(d) 2

5 2( ) ; 4

9 20

xf x a

x x

2. Sketch the graphs of the following functions and determine whether they are continuous on the closed interval 0,1 :

(a)

1, 0

( ) 0, 0 1

1, 1

x

f x x

x

(b)

1, 0

( )

1, 0

xf x x

x

(c)

2

2

, 0( )

, 0

x xf x

x x

(d) ( ) 1, 0 1f x x (e)

, 0

( ) 0, 0 1

, 1

x x

f x x

x x

Answer:

1. (a) Yes,5

lim ( ) (5)x

f x f

(b) No, (1)f undefined (c) No, 1

lim ( ) 2x

f x

but (1) 3f (d) No, (4)f undefined

2. (a) Yes; (b) No. Not continuous on the right at 0; (c) Yes; (d) No. Not defined at 0; (e) No. Not continuous on the left at 1