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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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
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
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.