AAT-A IB - HR Date: 3/11/2014 ID Check•Obj: SWBAT perform function operations.Bell Ringer: ACT Prep ProbsHW Requests: pg 375 #42, 43; Worksheet on finding roots of Polynomials #1-13 odds, 2, 4, 6 complete the factoring worksheets; pg 387 #17-22, 47, 48 HW: pg 386 #29-34 Read Section 7.8Announcements:“There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls.” ― Howard Thurman

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AAT-A IB - HR Date: 3/12/2014 ID Check•Obj: SWBAT perform function operations.Bell Ringer: ACT Prep ProbsHW Requests: pg 375 #42, 43; Worksheet on finding roots of Polynomials #1-13 odds, 2, 4, 6 complete the factoring worksheets; HW: pg 387 #23, 25,27,35-45odds#49-51, 68Read Section 7.8Announcements:“There is something in every one of you that waits and listens for the sound of the genuine in yourself. It is the only true guide you will ever have. And if you cannot hear it, you will all of your life spend your days on the ends of strings that somebody else pulls.” ― Howard Thurman

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COMPOSITION

OFFUNCTIONS

“SUBSTITUTING ONE FUNCTION INTO ANOTHER”

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

1

xgf

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.

1 is ofdomain xxgf

Example – Composition of Functions

xfgxfg

2)2()2( xxgxg

49)7( 2

5 Find . and 2xfLet 2 fgxxgx Method 1:

2255 fg

Method 2:

xfgxfg

)25(5 gfg

49)7( 2

)7(g

Let’s try some 2 Find .7 and xfLet 23 fgxxgx

Solution 2 Find .7 and xfLet 23 fgxxgx

Solving with a Graphing Calculator

2 Find .7 and xfLet 23 fgxxgx

Start with the y= list.

Input x3 for Y1 and x2+7 for Y2

Now go back to the home screen.

Press VARS, YVARS and select 1. You will get the list of functions.

Using VARS and YVARS enter the function as Y2(Y1(2).

You should get 71 as a solution.

Real Life Application

• You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item.

1. Write functions for the two situations. Let x = original price.

– 20% discount: f(x) = x – 0.20x = 0.8x– Cost with the coupon: g(x) = x - 5

You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item.

2. Make a composition of functions:

This represents if the clerk does the discount first, then takes $5 off the discounted price.

58.0

))8.0((

x

xgxfg

You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item.

3. Now try applying the $5 coupon first, then taking 20% off:

How much more will it be if the clerk applies the coupon BEFORE the discount?

4-0.8x

)5(8.0

))5(((

x

xgfxgf

The Composition Function

xgfxgf This is read “f composition g” or “f of g(x)” 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

FOIL first and then distribute

the 2

xfgxfg This is read “g composition f” or “g of f(x)” 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” or “f of f(x)” 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

Composite Function – When you combine two or more functions

• The composition of function g with function is written as xfgxfg

1

21. Evaluate the inner function f(x) first.

2. Then use your answer as the input of the outer function g(x).

You are shopping in a store that is offering 20% off everything. You also have a coupon for $5 off any item.

4. Subtract the two functions:

Any item will be $1 more if the coupon is applied first. You will save $1 if you take the discount, then use the coupon.

1)58.0()48.0(

xx

xfgxgf

7.6 Function Operations

Review: What is a function?

• A relationship where every domain (x value) has exactly one unique range (y value).

• Sometimes we talk about a FUNCTION MACHINE, where a rule is applied to each input of x

Function Operations

xgxfxgf )( :Addition

xgxfxgf :tionMultiplica

xgxfxgf :nSubtractio

0xg where :Division

xg

xfx

g

f

Adding and Subtracting Functions

45

)122()83(

)(

x

xx

xgxfxgf

20

)122()83(

)(

x

xx

xgxfxgf

g - f and g f Find

.122g and 83fLet

xxxx

When we look at functions we also want to look at their domains (valid x values). In this case, the domain is all real numbers.

Multiplying Functions

1

)1)(1()(23

2

xxx

xxxgxf

g f Find

.1g and 1-fLet 2

xxxx

In this case, the domain is all real numbers because there are no values that will make the function invalid.

Dividing Functions

1)1(

)1)(1(

1

12

xx

xx

x

x

xg

xf

g

f Find

.1g and 1-fLet 2 xxxx

In this case, the domain is all real numbers EXCEPT -1, because x=-1 would give a zero in the denominator.

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 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

xgxfxgf 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

1432 32 xxgf

31228 325 xxx

FOIL

Good idea to put in 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

fNothing more you could do here. (If you can reduce

these you should).

What is the domain?

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.

Let’s Try Some

)( Find xgxf )( Find xgxf

.15g and 1-5fLet 2 xxxx

What is the domain?

Let’s Try Some

)( Find xgxf )( Find xgxf

.15g and 1-5fLet 2 xxxx

What is the domain?

Let’s Try Some

)( Find xgxf g(x)

Findxf

.12g and 5-7x6fLet 2 xxxx

What is the domain?

Let’s Try Some

)( Find xgxf g(x)

Findxf

.12g and 5-7x6fLet 2 xxxx

What is the domain?