Objective To perform operations on functions and to determine the domains of the resulting...

Objective To perform operations on functions and to determine the domains of the resulting functions.

In real life, we need consider the profit function in terms of the item we made and sold. In fact, the profit function is the difference between the sales income and the cost in terms of the item we made and sold. If let x denote the number of item we made and sold, then the profit function can be expressed as

P(x) = S(x) – C(x)

In real life example, we can see that we do have the demand to consider the difference between the two functions

The sum f + g

xgxfxgf This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.

1432 32 xxgxxf

1432 32 xxgf

424 23 xx

Combine like terms & put in

power descending

order

The difference f - g

xgxfxgf To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms.

1432 32 xxgxxf

1432 32 xxgf

1432 32 xx

Distribute negative

224 23 xx

The product f • g

f g x f x g x To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function.

1432 32 xxgxxf

2 32 3 4 1f g x x

31228 325 xxx

Distributive Property

Good idea to put in power descending

order but not required.

The quotient f /g

xgxf

xg

f

To find the quotient of two functions, put the first one over the second.

1432 32 xxgxxf

14

323

2

x

x

g

f Nothing more you could do here. (If you can reduce

these you should).

21 1f x x and g x x

.a f g x .f

b xg

21 1x x 2x x 2

1

1

x

x

1

1 1

x

x x

1 , 1

1x x

x

R

Note that the domain of the resulting functions of the first 4 operations is determined at the set-up time, not the simplified time.

Example 1 Let.

Find the resulting function and domain.

xR

So the first 4 operations on functions are pretty straight forward.

The rules for the domain of functions would apply to these combinations of functions as well. The domain of the sum, difference or product would be the numbers x in the domains of both f and g.

For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0.

In one word, the domain of the resulting functions of the first 4 operations is determined at the set-up time, not the simplified time.

COMPOSITION

OFFUNCTIONS

“SUBSTITUTING ONE FUNCTION INTO ANOTHER”

The Composition Function

xgfxgf This is read “f composition g” and means to copy the f function down but where ever you see an x, substitute in the g function.

1432 32 xxgxxf

314223 xgf

51632321632 3636 xxxx

Square first and then distribute

the 2

xfgxfg This is read “g composition f” and means to copy the g function down but where ever you see an x, substitute in the f function.

1432 32 xxgxxf

132432 xfg

You could multiply this out but since it’s to the 3rd power we

won’t

xffxff This is read “f composition f” and means to copy the f function down but where ever you see an x, substitute in the f function. (So sub the function into itself).

1432 32 xxgxxf

332222 xff

The DOMAIN of the Composition Function

The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

11

xxgx

xf

1( )

1f g x

x

The domain of g is x 1

We also have to worry about any “illegals” in this composition function, specifically dividing by 0. This would mean that x 1 so the domain of the composition would be combining the two restrictions.

domain of is | , 1f g x x x

| , 0y y y | , 6x x x

The DOMAIN and RANGE of Composite Functions

We could first look at the domain and range of f(x) and g(x).

1

15

x

xgxxf

Hence we must exclude 6 from the domain of f(x)

For g(x) to cope with the output from f(x) we must ensure that the output of f(x) does not include 1

5xxf

?)( xfg

| , 1y y y

1

1

x

xg

1 x

| , 0y y y | , 6x x x


1

15

x

xgxxf

Hence we must exclude 6 from the

domain of f(x)

5xxf

?)( xfg

1

1

x

xg

1( )

5 1g f x

x

1

6x

Or we could find g o f (x) and determine the domain and range of the resulting expression.

| , 5y y y | , 1x x x



1

15

x

xgxxf

Hence we must exclude 1 from the domain of g(x)

For f(x) to cope with the output from g(x) we must ensure that the output does not include 0

1

1

x

xg

?)( xgf

| , 0y y y

5xxf

0 x

| , 5y y y | , 1x x x


Or we could find f o g (x) and determine the domain and range of the resulting expression.

1

15

x

xgxxf

51

1)(

x

xgf

Domain: Range:

0: yy

0: xx



2xxgxxf

xxf

?)( xfg

0: yy

2xxg

0 x

0: yy 0: xx



2xxgxxf

However, the approach that finding domain and range directly from the expression of the composition must be used with CAUTION.

xxfg )(

Domain: Range:

yandx

Note that the domain of the resulting functions is determined at the set-up time, not the simplified time. Not:

| , 0y y y | , 2x x x



22 xxgxxf

f o g (x) is a function for the natural domain of g(x)

f(x) can cope with all the numbers in the range of g(x) because the range of g(x) is contained within the domain of f(x)

2 xxg

?)( xgf

| , 0y y y

xxf 2

0 x

0: yy



22 xxgxxf

g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

xxf 2

?)( xfg

y

2 xxg

x

| , 0y y y | , 1x x x



22 xxgxxfgof (x) is not a function for the domain of g(x) unless we restrict

the domain of f(x)

g(x) cannot cope with all the numbers in the range of f(x). Need to restrict the domain f(x) to give an output that g(x) can cope with.

xxf 2

?)( xfg

y

2 xxg

x | 2x x | 2y y

| 2 2x x

Example 2. Let 2

12f x and g x x

x

Find a rule for f g x And give the domain of the composite function.

2f g x f g x f x

2

1

2x

1

2x

The domain of 2f g is x x even though the expression1

2x is also defined for x < 2.

Example 3. Let 11f x and g x x

x

Find rules for f g x and g f x and give the domain of each composite function.

11 1

1f g x f g x f x domain x x

x

1 11 0g f x g f x g domain x x

x x

A Challenge Example of Composite Functions

14132))((5 2 xxxfgxxf

?)( xg

Try it !!

???50202

???)5(2)5(2

2

xx

xxg

25 2 13 14f x x g f x x x

Method 1

Complete the square with (x – 5)

2

2

( ) 2 13 14

( 5) 2 13 14

g f x x x

g x x x

2( 5) 2( 5) 7 36g x x x 22( 5) 7 35 1x x 22( 5) 7( 5) 1x x

25 2 13 14f x x g f x x x

Method 1

2( ) 2 7 1g x x x

2( 5) 2( 5) 7( 5) 1g x x x 2( ( )) 2( ( )) 7( ( )) 1g f x f x f x

5 yx

25 2 13 14f x x g f x x x

5)( xxfyLet

Method 2

2

2

( ) 2 13 14

( ) 2 13 14

g f x x x

g y x x

172

14651350202

14)5(13)5(2)(

2

2

2

yy

yyy

yyyg

25 2 13 14f x x g f x x x

Method 2

172)( 2 xxxg

172

14651350202

14)5(13)5(2)(

2

2

2

yy

yyy

yyyg

Of the Method 1 and Method 2, the Method 2 is BETTER!!!

Assignment

P. 123 #18; P. 128 #1 – 25 (odd); WS 4.1

| , 0y y y | , 6x x x



1

15

x

xgxxf

However this approach must be used with CAUTION.

6

1)(

x

xfg

Domain: Range:

Objective To perform operations on functions and to determine the domains of the resulting...

Documents

Transcript of Objective To perform operations on functions and to determine the domains of the resulting...