evaluate composition of functions
TRANSCRIPT
March 24, 2014
Evaluatefor x = 4
for t = -2
for x = 6
Composition of Functions
When you plug a function into a second function, you are doing composition of functions.
Plug g(x) into f(x)
Plug f(x) into g(x)
ALWAYS work from the INSIDE to the OUTSIDE!
Example: Find f(g(x)) and g(f(x)). Example: Find and .
March 24, 2014
Evaluate. Inverse of a Relation
The inverse of a relation consisting of the ordered pairs (x, y)
is the set of all ordered pairs (y, x).
Notation:
Represents the inverse of the function
One-to-One Definition:A function is one-to-one, if there is exactly one x for every y value (in addition to there being exactly one y for every x).
Example of a function that is one-to-one. Example of a function that is NOT one-to-one.
2345
6789
2345
6
78
x y x y
If the inverse of a function is a function, then the function is one-to-one.
Find the inverse of each function. State whether the function is one-to-one.
a. {(5, 2), (4, 3), (3, 4), (2, 5)}
c. {(1, 2), (4, 3), (2, −1), (5, 3)}
March 24, 2014
Horizontal-Line Test
The inverse of a function is a function if and only if every horizontal line intersects the graph of the given function (passed the vertical-line test) at no more than one point.
If a function passes both the vertical line test AND the horizontal line test, then it is a one-to-one function.
Determine whether the function is one-to-one.
Inverse Functions
Function:
Inverse Function:
To find the inverse equation of a function
1. Change f(x) to y.
2. Interchange x and y
3. Solve for y
4. Change new y to f-1(x)
March 24, 2014
Finding Inverses
1. 2.
Find the inverse of each function.
1.
2.
You tryFind the inverse of the following functions
1.
2.
The graph of a function and its inverse is symmetrical with respect to the y = x line.
This is called the identity function.
March 24, 2014
Graph the inverse of the graph. (Use y=x to find inverse points)
Inverses give you back the original value
Examples
1.
2.
if x=4, then f(4) = 2(4)+3 = 11
We can verify that two functions are inverses of each other by determining if the composition of the two functions are both equal to x.
Use composition to determine if the following functions are inverses of each other.
a) b)
March 24, 2014
Use composition to determine if the following functions are inverses of each other.
a) b)
Find the inverse algebraically then graph it.
xx f(x) f(x)
Solve for the inverse algebraically then graph it.
xx f(x) f(x)
Check for understanding
#1 Graph the function
#2 Write an equation for the graph
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**************
**State*the*Domain*of*the*function____________________________________________________________________**State*the*Range*of*the*function______________________________________________________________________** * * * * * * * * * * *! ! = !!* * Inverse* * ** * * * ! ! = !! * * * ** *
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*Domain:_______________________________**Range:________________________________**Describe*why*the*domain*is*different*for*the*two*parent*graphs.***
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Examples*of*a*vertical*shift.***! ! = ! + 2*** ! ! = ! − 2* * * ! ! = ! + 1* !** *****Discuss*with*a*partner*any*patterns*you*may*see.*Predict*what*the*graph*will*look*like*for*the*following*function.**Sketch*your*prediction*below.**! ! = ! − 1* * * * * * * *Use*your*graphing*calculator**
to*check*your*prediction.***
*********Examples*of*a*horizontal*shift.!**! ! = ! + 2* * * * ******! ! = ! − 2* * * ********! ! = ! + 1* ** *** *
Discuss*with*a*partner*any*patterns*you*may*have*noticed*from*the*previous*3*examples.**Predict*what*the*graph*will*look*like*for*the*following*function.**Sketch*your*prediction*on*the*given*graph*below.***! ! = ! − 1* * * * * * *Use*your*graphing*calculator**
*to*check*your*prediction.*** * * * * * * * ** * * * * * * * ********Examples*of*a*vertical*stretch****! ! = 2 !* * * * ***! ! = 3 !* * * * * ! ! = 5 !* ** * * * **Discuss*with*a*partner*any*patterns*you*many*have*noticed*from*the*examples*above.**Predict*what*the*graph*will*look*like*for*the*following*function.!**! ! = 4 !**
***Use*a*graphing*calculator**to*check*your*solution.*
Examples*of*a*vertical*stretch*and*flip.***! ! = −2 !** * * * * * ! ! = −3 !* *** ************Discuss*with*a*partner*any*patterns*you*many*have*noticed*from*the*examples*above.**Predict*what*the*graph*will*look*like*for*the*following*function.!**! ! = −5 !********
*Use*a*graphing*calculator**to*check*your*solution.*
*********Use*the*information*you*have*gathered*from*all*of*the*examples*and*predict*and*sketch*the*following*function.**Check*your*answer*with*your*calculator.***
! ! = ! + 5− 2*************
Use!the!parent!graph!of! !! !to!create!example!problems!of!the!different!types!of!transformations.!!!*Example*problems*of*a*vertical*shift*.****______________________________________________* * * ______________________________________________* ** * * * * *(Write*function*above)*(Graph*function*below)*******************Example*problems*of*a*horizonal*shift*.****______________________________________________* * * ______________________________________________* ** * * * * *(Write*function*above)*(Graph*function*below)*******************
Example*problems*of*a*vertical*stretch*****______________________________________________* * * ______________________________________________* ** * * * * *(Write*function*above)*(Graph*function*below)*****************Example*problems*of*a*vertical*stretch*and*flip***______________________________________________* * * ______________________________________________* ** * * * * *(Write*function*above)*(Graph*function*below)*********
Name: ___________________________ Date: ___________ Period: ________ Secondary Math II 11-1 In-Class
Evaluating*and*Composing*Functions!Evaluate*for*x*=*3.**
1.!! 2( ) 5h x x= + !! ! ! ! ! 2.!! 6( )j xx
= !!!!!*Find* ( )(2)f go .*
3.!!4( ) 6
( )
f xx
g x x
= −
=!! ! ! 4.!!
( ) 3
( )2
f x xx
g x
= −
=!! ! ! 5.!!
2( )
( )
f x x
g x x
=
=!!
!!!!!Find ( )(2)g fo .*
6.!! ( ) 5( ) 3f x xg x x
= −=
! ! ! 7.!! ( ) 5( ) 2 3f x xg x x
= += − +
! ! ! 8.!!2f( )3g( ) 12
x x
x x
=
= −!
!!!!!!Find*f(g(x))*and*g(f(x)).*
9.!! ( ) 53
( ) 6 9
xf x
g x x
= − +
= −!!
!!!!!!!LOOKING*AHEAD:**Find* ( )( )f g xo *and* ( )( )g f xo *.**What*do*you*notice*about*the*new*functions?**Why*do*you*think*this*happens?**
10.!!( ) 2 10
1( ) 52
f x x
g x x
= −
= +**
!
Name: ___________________________ Date: ___________ Period: ________ Secondary Math II 11-2 In-Class
Inverses!Write&the&inverse&of&the&function.&Determine&if&the&function&is&one2to2one.&1.#{ }(3,4),(4,3),(7,5),(5,7) ## # # # ##&Determine&whether&the&function&is&one2to2one.&2.# # # ######## # # # # 3.# # # # ####&&&&&&For&each&function,&find&an&equation&of&the&inverse.&&&
4.## (x) 2x 7f = − − ## # # 5.### 3(x)
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f−= # # # 6.### 3( ) 4g x x= − #
#####Verify,&using&composition,&that&the&following&are&inverses.&
7.&&1( )4
( ) 4 1
xf x
g x x
−=
= +## # # 8.###
3( ) 22
2 3( )4
f x x
xg x
= +
−=## # # 9.###
1( ) ( 6)
3( ) 3x 6
f x x
g x
= −
= +##
# # # # #########Find&the&inverse&of&each&function.&&Then&verify,&using&composition&that&they&are&inverses&of&each&other.&
10.#### f (x) = x −1
3## # # # # ##
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Name: ___________________________ Date: ___________ Period: ________ Secondary Math II 11-3 In-Class
Root$Functions!Write the!function!from!the!given!graph.!!1.!!! ! ! ! ! ! ! ! 2.!! ! ! !!!!!!!!!!!Function:! ! ! ! ! ! ! ! Function:!!!3.! ! ! ! ! ! ! ! 4.!!! ! ! !!!!!!Sketch!the!graph!of!the!given!function.!!5.!! f (x) = x + 5 ! ! ! ! ! ! 6.!!(!) = ! + 1! − 5!!!! !!!!!!!!!!!!
!!7. f (x) = 3 x + 2 ! ! ! ! ! ! 8.!!(!) = ! − 2+ 3!!!!!!!!!!!!!!!!!9.!!!(!) = − ! + 5! ! ! ! ! ! 10.!! f (x) = 2 x3 !!!