Composite and Inverse Functions · Composite Functions •In a composition of functions, the range...
Transcript of Composite and Inverse Functions · Composite Functions •In a composition of functions, the range...
![Page 1: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/1.jpg)
Composite and Inverse Functions
![Page 2: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/2.jpg)
3rd
![Page 3: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/3.jpg)
4th Block
![Page 4: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/4.jpg)
Composite Functions
• In a composition of functions, the range of one function is part of the domain of the other function
• Basically substituting one function into another function
• Notation of composite functions is
• f(g(x)) or (f◦g)(x)
• Read as “f of g of x”
![Page 5: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/5.jpg)
Composite Functions
)2)(())((
))(())((
...
4)(
12)(34)( 2
fhgxhg
xfgxgf
find
xxh
xxgandxxxf
![Page 6: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/6.jpg)
Composite Functions
)1)(())((
))(())((
...
)(
13)(2)( 2
fhxhg
xfgxgf
find
xxh
xxgandxxxf
![Page 7: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/7.jpg)
Inverse Functions
• The domain of the inverse is the range of the original function
• The range of the inverse is the domain of the original function
• Switching x and y
![Page 8: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/8.jpg)
To determine whether the inverse is a function…
• Switch x and y values and determine whether the domain of inverse is paired with only one value in the range (domain can not repeat)
:Inverse
(3,3) (-3,3) (5,2) (4,2) (2,3) :Relation
function a isit whether decide andrelation theof inverse theDetermine
![Page 9: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/9.jpg)
Determining whether the inverse is a function from the graph of the original
• You do the vertical line test to determine whether the graph represents a function
• To determine whether its inverse would be a function, then you would do a horizontal line test
![Page 10: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/10.jpg)
Finding the Inverse of a Relation
What is the inverse of relation s?
Relation s
X Y
0 -1
2 0
3 2
4 3
![Page 11: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/11.jpg)
To find the inverse of a function
• Notation: f -1(x)=
• Steps:
1. Replace f(x) with y
2. Switch the x and y variable
3. Solve for y
4. Replace y with f -1(x)=
![Page 12: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/12.jpg)
Find the following
)2(
)()(
)()(
3
2)(1)(
25)(32)(
1
11
11
2
f
xmxh
xgxf
xxmxxh
xxgxxf
![Page 13: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/13.jpg)
Find the following
)3(
)()(
)()(
5)(2)(
1)(14)(
1
11
11
2
f
xmxh
xgxf
xxmxxh
xxgxxf
![Page 14: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/14.jpg)
Inverse functions
?
25
1)(105)(
))(())((
inversestheyare
xxgxxf
inversesarefunctionstwothethen
xxfgandxxgfif
![Page 15: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/15.jpg)
Are They Inverses?
• 𝑓 𝑥 = 4𝑥 + 6
• 𝑔 𝑥 =𝑥−6
4
![Page 16: Composite and Inverse Functions · Composite Functions •In a composition of functions, the range of one function is part of the domain of the other function •Basically substituting](https://reader030.fdocuments.in/reader030/viewer/2022040413/5f0bef467e708231d432f12d/html5/thumbnails/16.jpg)
Are They Inverses?
• 𝑓 𝑥 = −6𝑥
• 𝑔 𝑥 =1
6𝑥