7.6 function operations. review: what is a function? a relationship where every domain (x value has...
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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
g - f and g f Find
.122g and 83fLet
xxxx
20
)122()83(
)(
x
xx
xgxfxgf
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.
Let’s Try Some
)( Find xgxf
.15g and 1-5fLet 2 xxxx
)( Find xgxf What is the domain?
Let’s Try Some
)( Find xgxf
.15g and 1-5fLet 2 xxxx
)( Find xgxf What is the domain?
Let’s Try Some
)( Find xgxf
.12g and 5-7x6fLet 2 xxxx
g(x) FindxfWhat is the domain?
Let’s Try Some
)( Find xgxf
.12g and 5-7x6fLet 2 xxxx
g(x) FindxfWhat is the domain?
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).
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 they 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
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