11 x1 t02 08 inverse functions (2010)
TRANSCRIPT
![Page 1: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/1.jpg)
Inverse Relations
![Page 2: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/2.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x 3inverse relation is x y y
![Page 3: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/3.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x
The domain of the relation is the range of its inverse relation
3inverse relation is x y y
![Page 4: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/4.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x
The domain of the relation is the range of its inverse relation
3inverse relation is x y y
The range of the relation is the domain of its inverse relation
![Page 5: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/5.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x
The domain of the relation is the range of its inverse relation
A relation and its inverse relation are reflections of each other in the line y = x.
3inverse relation is x y y
The range of the relation is the domain of its inverse relation
![Page 6: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/6.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x
The domain of the relation is the range of its inverse relation
A relation and its inverse relation are reflections of each other in the line y = x.
2e.g. y x
3inverse relation is x y y
The range of the relation is the domain of its inverse relation
![Page 7: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/7.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x
The domain of the relation is the range of its inverse relation
A relation and its inverse relation are reflections of each other in the line y = x.
2e.g. y x
3inverse relation is x y y
The range of the relation is the domain of its inverse relation
domain: all real xrange: 0y
![Page 8: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/8.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x
The domain of the relation is the range of its inverse relation
A relation and its inverse relation are reflections of each other in the line y = x.
2e.g. y x
3inverse relation is x y y
The range of the relation is the domain of its inverse relation
domain: all real xrange: 0y
2inverse relation: x y
![Page 9: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/9.jpg)
Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained by interchanging x and y is x = f(y)
3e.g. y x x
The domain of the relation is the range of its inverse relation
A relation and its inverse relation are reflections of each other in the line y = x.
2e.g. y x
3inverse relation is x y y
The range of the relation is the domain of its inverse relation
domain: all real xrange: 0y
2inverse relation: x ydomain: 0x
range: all real y
![Page 10: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/10.jpg)
Inverse Functions
![Page 11: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/11.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
![Page 12: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/12.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test
![Page 13: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/13.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx
![Page 14: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/14.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
![Page 15: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/15.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
![Page 16: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/16.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
![Page 17: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/17.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
OR2yx
xy
![Page 18: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/18.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
OR2yx
xy NOT UNIQUE
![Page 19: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/19.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
OR2yx
xy NOT UNIQUE
y
x
3xyii
![Page 20: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/20.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
OR2yx
xy NOT UNIQUE
y
x
3xyii
![Page 21: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/21.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
OR2yx
xy NOT UNIQUE
y
x
3xyii
Has an inverse function
![Page 22: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/22.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
OR2yx
xy NOT UNIQUE
y
x
3xyii
Has an inverse function
OR3yx
3 xy
![Page 23: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/23.jpg)
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
Testing For Inverse Functions(1) Use a horizontal line test OR unique.is ,asrewritten is When 2 xgyxgyyfx 2xyi y
x
Only has an inverse relation
OR2yx
xy NOT UNIQUE
y
x
3xyii
Has an inverse function
OR3yx
3 xy UNIQUE
![Page 24: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/24.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
![Page 25: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/25.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
e.g. 2
2xf xx
![Page 26: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/26.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
e.g. 2
2xf xx
2 22 2
x yy xx y
![Page 27: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/27.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
e.g. 2
2xf xx
2 22 2
x yy xx y
2 22 2
1 2 22 21
y x yxy x yx y x
xyx
![Page 28: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/28.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
e.g. 2
2xf xx
2 22 2
x yy xx y
2 22 2
1 2 22 21
y x yxy x yx y x
xyx
1
22 22
212
xxf f x
xx
![Page 29: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/29.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
e.g. 2
2xf xx
2 22 2
x yy xx y
2 22 2
1 2 22 21
y x yxy x yx y x
xyx
1
22 22
212
xxf f x
xx
2 4 2 42 2
44
x xx xx
x
![Page 30: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/30.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
e.g. 2
2xf xx
2 22 2
x yy xx y
2 22 2
1 2 22 21
y x yxy x yx y x
xyx
1
22 22
212
xxf f x
xx
2 4 2 42 2
44
x xx xx
x
1
2 2 21
2 2 21
xxf f x
xx
![Page 31: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/31.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1 AND xxff 1
e.g. 2
2xf xx
2 22 2
x yy xx y
2 22 2
1 2 22 21
y x yxy x yx y x
xyx
1
22 22
212
xxf f x
xx
2 4 2 42 2
44
x xx xx
x
1
2 2 21
2 2 21
xxf f x
xx
2 2 2 22 2 2 244
x xx xx
x
![Page 32: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/32.jpg)
(ii) Draw the inverse relation
y
x
![Page 33: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/33.jpg)
(ii) Draw the inverse relation
y
x
![Page 34: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/34.jpg)
(ii) Draw the inverse relation
y
x
![Page 35: 11 X1 T02 08 inverse functions (2010)](https://reader034.vdocuments.us/reader034/viewer/2022042714/5559c7a8d8b42a236c8b56b8/html5/thumbnails/35.jpg)
(ii) Draw the inverse relation
y
x
Exercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj