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Combining Functions: Addition, Subtraction, Multiplication, Division, Composition, Study notes of Mathematics

Capitol University (CU)Mathematics

A section from a university mathematics textbook that covers the concepts of combining functions through addition, subtraction, multiplication, division, and composition. It includes definitions, examples, and exercises for students to practice. from the University of Houston Department of Mathematics Precalculus course (MATH 1330).

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SECTION 1.4 Operations on Functions

MATH 1330 Precalculus

107

Section 1.4: Operations on Functions

 Combining Functions by Addition, Subtraction, Multiplication,

Division, and Composition

Combining Functions by Addition, Subtraction,

Multiplication, Division, and Composition

Definition of the Sum, Difference, Product, Quotient, and

Composition of Functions:

Sum:

Difference:

Partial preview of the text

Download Combining Functions: Addition, Subtraction, Multiplication, Division, Composition and more Study notes Mathematics in PDF only on Docsity!

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 107

Section 1.4 : Operations on Functions

 Combining Functions by Addition, Subtraction, Multiplication,

Division, and Composition

Combining Functions by Addition, Subtraction,

Multiplication, Division, and Composition

Definition of the Sum, Difference, Product, Quotient, and

Composition of Functions:

Sum:

Difference:

CHAPTER 1 A Review of Functions

108 University of Houston Department of Mathematics

Product:

Quotient:

Composition:

Example:

Solution:

CHAPTER 1 A Review of Functions

110 University of Houston Department of Mathematics

Example:

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 111

Additional Example 1:

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 113

Additional Example 2:

CHAPTER 1 A Review of Functions

114 University of Houston Department of Mathematics

Solution:

Additional Example 3:

CHAPTER 1 A Review of Functions

116 University of Houston Department of Mathematics

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 117

Additional Example 5:

Solution:

Exercise Set 1.4: Operations on Functions

MATH 1330 Precalculus 119

       

















       





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



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

Answer the following.

(a) Find f (  3 ) g ( 3 ).

(b) Find f ( 0 ) g ( 0 ).

(c) Find f (  6 ) g ( 6 ).

(d) Find f ( 5 ) g ( 5 ).

(e) Find f ( 7 ) g ( 7 ).

(f) Sketch the graph of f  g. (Hint: For any x

value, add the y values of f and g .)

(g) What is the domain of f  g? Explain how

you obtained your answer.

(a) Find f (  2 ) g ( 2 ).

(b) Find f ( 0 ) g ( 0 ).

(c) Find f (  4 ) g ( 4 ).

(d) Find f ( 2 ) g ( 2 ).

(e) Find f ( 4 ) g ( 4 ).

(f) Sketch the graph of f  g. (Hint: For any x

value, subtract the y values of f and g .)

(g) What is the domain of f  g? Explain how

you obtained your answer.

For each of the following problems:

(f) Find f  g and its domain.

(g) Find f  g and its domain.

(h) Find fg and its domain.

(i) Find

f

g

and its domain.

Note for (a)-(d): Do not sketch any graphs.

2 fx  x  gx  x  x 

3 2 fx  x  x gx  x  x 

x gx x

7. f ( x ) x  6 ; g ( x ) 10  x

8. f ( x ) 2 x  3 ; g ( x ) x  4

2 2 fx  x  gx  x 

2 fx   x gx  x 

Find the domain of each of the following functions.

 x x

h x x

x x

x fx

x f x

x gx

Exercise Set 1.4: Operations on Functions

120 University of Houston Department of Mathematics

Answer the following, using the graph below.

17. (a) g ( 2 ) (b) f  g   2 

(c) f ( 2 ) (d) g  f   2 

18. (a) g ( 0 ) (b) f  g   0 

(c) f ( 0 ) (d) g  f   0 

19. (a)  f  g   3  (b)  g  f   3 

20. (a)  f  g   1  (b)  g  f   1 

21. (a)  f  f   3 (b)  g  g  2 

22. (a)  f  f   5 (b)  g  g  3 

23. (a)  f  g   4 (b)  g  f   4

24. (a)  f  g   5  (b)  f  g   2

Use the functions f and g given below to evaluate the

following expressions:

( ) 3 2 and () 5 4

2 fx   x gx  x  x 

25. (a) g ( 0 ) (b) f  g   0 

(c) f ( 0 ) (d) g  f   0 

26. (a) g ( 1 ) (b) f  g  1 

(c) f ( 1 ) (d) g  f  1 

27. (a)  f  g   2  (b)  g  f   2 

28. (a)  f  g   4 (b)  g  f   4

29. (a)  f  f   6 (b)  g  g   6 30. (a) (^)  f  f   4  (b)  g  g  4 

31. (a)  f  g   x (b)  g  f   x

32. (a)  f  f   x (b)  g  g   x

The following method can be used to find the domain

of f g :

(a) Find the domain of g.

(b) Find f g.

(c) Look at the answer from part (b) as a stand-

alone function (ignoring the fact that it is a

composition of functions) and find its domain.

(d) Take the intersection of the domains found in

steps (a) and (c). This is the domain of f g.

Note:

We check the domain of g because it is the inner

function of (^) f g, i.e. f (^)  g (^)  x . If an x-value is

not in the domain of g, then it also can not be an

input value for f g.

Use the above steps to find the domain of f g for the

following problems:

f ( ) x ; g x ( ) x 5

x

f ( ) x ; g x ( ) x 2

x

f x g x x

x

f x g x x

x

For each of the following problems:

(a) Find f g and its domain.

(b) Find g f and its domain.

2 fx  x  x gx  x 

2 f ( ) x  6 x  2; g x ( )  7  x

fx x gx

2 ; ( ) 5

( ) gx x x

f x  

       

















Combining Functions: Addition, Subtraction, Multiplication, Division, Composition, Study notes of Mathematics

Related documents

Partial preview of the text

Download Combining Functions: Addition, Subtraction, Multiplication, Division, Composition and more Study notes Mathematics in PDF only on Docsity!

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 107

Section 1.4 : Operations on Functions

 Combining Functions by Addition, Subtraction, Multiplication,

Division, and Composition

Combining Functions by Addition, Subtraction,

Multiplication, Division, and Composition

Definition of the Sum, Difference, Product, Quotient, and

Composition of Functions:

Sum:

Difference:

CHAPTER 1 A Review of Functions

108 University of Houston Department of Mathematics

Example:

Solution:

CHAPTER 1 A Review of Functions

110 University of Houston Department of Mathematics

Example:

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 111

Additional Example 1:

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 113

Additional Example 2:

CHAPTER 1 A Review of Functions

114 University of Houston Department of Mathematics

Solution:

Additional Example 3:

CHAPTER 1 A Review of Functions

116 University of Houston Department of Mathematics

Solution:

SECTION 1.4 Operations on Functions

MATH 1330 Precalculus 117

Additional Example 5:

Solution:

MATH 1330 Precalculus 119

Answer the following.

(a) Find f (  3 ) g ( 3 ).

(b) Find f ( 0 ) g ( 0 ).

(c) Find f (  6 ) g ( 6 ).

(d) Find f ( 5 ) g ( 5 ).

(e) Find f ( 7 ) g ( 7 ).

(f) Sketch the graph of f  g. (Hint: For any x

value, add the y values of f and g .)

(g) What is the domain of f  g? Explain how

you obtained your answer.

(a) Find f (  2 ) g ( 2 ).

(b) Find f ( 0 ) g ( 0 ).

(c) Find f (  4 ) g ( 4 ).

(d) Find f ( 2 ) g ( 2 ).

(e) Find f ( 4 ) g ( 4 ).

(f) Sketch the graph of f  g. (Hint: For any x

value, subtract the y values of f and g .)

(g) What is the domain of f  g? Explain how

you obtained your answer.

For each of the following problems:

(f) Find f  g and its domain.

(g) Find f  g and its domain.

(h) Find fg and its domain.

(i) Find

f

g

and its domain.

Note for (a)-(d): Do not sketch any graphs.

7. f ( x ) x  6 ; g ( x ) 10  x

8. f ( x ) 2 x  3 ; g ( x ) x  4

Find the domain of each of the following functions.

120 University of Houston Department of Mathematics

Answer the following, using the graph below.

17. (a) g ( 2 ) (b) f  g   2 

(c) f ( 2 ) (d) g  f   2 

18. (a) g ( 0 ) (b) f  g   0 

20. (a)  f  g   1  (b)  g  f   1 

22. (a)  f  f   5 (b)  g  g  3 