12 x1 t05 01 inverse functions (2013)
-
Upload
nigel-simmons -
Category
Education
-
view
671 -
download
1
Transcript of 12 x1 t05 01 inverse functions (2013)
![Page 1: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/1.jpg)
Inverse Functions
![Page 2: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/2.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
![Page 3: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/3.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)
![Page 4: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/4.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.
![Page 5: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/5.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.
If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1
![Page 6: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/6.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.
If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1
A function and its inverse function are reflections of each other in the line y = x.
![Page 7: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/7.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.
If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1
A function and its inverse function are reflections of each other in the line y = x.
xfyabxfyba 1on point a is , then ,on point a is , If
![Page 8: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/8.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.
If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1
A function and its inverse function are reflections of each other in the line y = x.
xfyabxfyba 1on point a is , then ,on point a is , If
xfyxfy 1 of range theis ofdomain The
![Page 9: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/9.jpg)
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
The relation obtained by interchanging x and y is x = f(y)yyxxxy 33 e.g.
If in this new relation, for each x value in the domain there is a maximum of one y value, (i.e. it is a function), then it is called the inverse function to y = f(x) and is symbolised xfy 1
A function and its inverse function are reflections of each other in the line y = x.
xfyabxfyba 1on point a is , then ,on point a is , If
xfyxfy 1 of range theis ofdomain The
xfyxfy 1 ofdomain theis of range The
![Page 10: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/10.jpg)
Testing For Inverse Functions
![Page 11: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/11.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
![Page 12: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/12.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi y
x
![Page 13: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/13.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi y
x
Only has an inverse relation
![Page 14: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/14.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation
![Page 15: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/15.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation Has an inverse function
![Page 16: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/16.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation Has an inverse function
OR unique.is ,asrewritten is When 2 xgyxgyyfx
![Page 17: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/17.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation Has an inverse function
OR unique.is ,asrewritten is When 2 xgyxgyyfx
2yx OR
![Page 18: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/18.jpg)
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation Has an inverse function
OR unique.is ,asrewritten is When 2 xgyxgyyfx
2yx OR
xy NOT UNIQUE
![Page 19: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/19.jpg)
Testing For Inverse Functions
(1) Use a horizontal line testOR
unique.is ,asrewritten is When 2 xgyxgyyfx e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation Has an inverse functionOR OR
2yx xy
NOT UNIQUE
3yx
![Page 20: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/20.jpg)
Testing For Inverse Functions
(1) Use a horizontal line testOR
unique.is ,asrewritten is When 2 xgyxgyyfx e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation Has an inverse functionOR OR
2yx xy
NOT UNIQUE
3yx 3 xy
UNIQUE
![Page 21: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/21.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
![Page 22: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/22.jpg)
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1
![Page 23: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/23.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 24: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/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
e.g.
xxxf
2312
![Page 25: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/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.
xxxf
2312
yyx
xxy
2312
2312
![Page 26: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/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.
xxxf
2312
yyx
xxy
2312
2312
2213132212231223
xxy
xyxyxyxyxy
![Page 27: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/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.
xxxf
2312
yyx
xxy
2312
2312
2213132212231223
xxy
xyxyxyxyxy
2
23122
123
1231
xx
xx
xff
![Page 28: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/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.
xxxf
2312
yyx
xxy
2312
2312
2213132212231223
xxy
xyxyxyxyxy
2
23122
123
1231
xx
xx
xff
x
xxxxx
88
46242336
![Page 29: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/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.
xxxf
2312
yyx
xxy
2312
2312
2213132212231223
xxy
xyxyxyxyxy
2
23122
123
1231
xx
xx
xff
x
xxxxx
88
46242336
221323
122132
1
xx
xx
xff
![Page 30: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/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.
xxxf
2312
yyx
xxy
2312
2312
2213132212231223
xxy
xyxyxyxyxy
2
23122
123
1231
xx
xx
xff
x
xxxxx
88
46242336
221323
122132
1
xx
xx
xff
x
xxxxx
88
26662226
![Page 31: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/31.jpg)
Restricting The Domain
![Page 32: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/32.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.
![Page 33: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/33.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.
![Page 34: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/34.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.
3 e.g. xyi y
x
3xy
![Page 35: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/35.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.
3 e.g. xyi y
x
3xy Domain: all real x
Range: all real y
![Page 36: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/36.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.
3 e.g. xyi y
x
3xy Domain: all real x
Range: all real y
31
31 :
xy
yxf
![Page 37: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/37.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.
3 e.g. xyi y
x
3xy Domain: all real x
Range: all real y
31
31 :
xy
yxf
Domain: all real x
Range: all real y
![Page 38: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/38.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.
3 e.g. xyi y
x
3xy Domain: all real x
Range: all real y
31
31 :
xy
yxf
Domain: all real x
Range: all real y
![Page 39: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/39.jpg)
Restricting The Domain
If a function does not have an inverse, we can obtain an inverse function by restricting the domain of the original function.When restricting the domain you need to capture as much of the range as possible.
3 e.g. xyi y
x
3xy
31
xy
Domain: all real x
Range: all real y
31
31 :
xy
yxf
Domain: all real x
Range: all real y
![Page 40: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/40.jpg)
xeyii y
x
xey
1
![Page 41: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/41.jpg)
xeyii y
xey
Domain: all real x
Range: y > 0 1
![Page 42: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/42.jpg)
xeyii y
x
xey
Domain: all real x
Range: y > 0
xyexf y
log:1
1
![Page 43: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/43.jpg)
xeyii y
x
xey
Domain: all real x
Range: y > 0
xyexf y
log:1
Domain: x > 0
Range: all real y
1
![Page 44: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/44.jpg)
xeyii y
x
xey
Domain: all real x
Range: y > 0
xyexf y
log:1
Domain: x > 0
Range: all real y
1
![Page 45: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/45.jpg)
xeyii y
x
xey
xy logDomain: all real x
Range: y > 0
xyexf y
log:1
Domain: x > 0
Range: all real y
1
1
![Page 46: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/46.jpg)
2xyiii y
x
2xy
![Page 47: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/47.jpg)
2xyiii y
x
2xy Domain: all real x
Range: 0y
![Page 48: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/48.jpg)
2xyiii y
x
2xy Domain: all real x
Range: 0y
NO INVERSE
![Page 49: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/49.jpg)
2xyiii y
x
2xy Domain: all real x
Range: 0y
NO INVERSERestricted Domain: 0x
![Page 50: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/50.jpg)
2xyiii y
x
2xy Domain: all real x
Range: 0y
NO INVERSERestricted Domain:
Range: 0y
0x
![Page 51: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/51.jpg)
2xyiii y
x
2xy Domain: all real x
Range:
21
21 :
xy
yxf
0y
NO INVERSERestricted Domain:
Range: 0y
0x
![Page 52: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/52.jpg)
2xyiii y
x
2xy Domain: all real x
Range:
21
21 :
xy
yxf
Domain:
Range:
0y
NO INVERSERestricted Domain:
Range: 0y
0x
0x0y
![Page 53: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/53.jpg)
2xyiii y
x
2xy Domain: all real x
Range:
21
21 :
xy
yxf
Domain:
Range:
0y
NO INVERSERestricted Domain:
Range: 0y
0x
0x0y
![Page 54: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/54.jpg)
2xyiii y
x
2xy 21
xy Domain: all real x
Range:
21
21 :
xy
yxf
Domain:
Range:
0y
NO INVERSERestricted Domain:
Range: 0y
0x
0x0y
![Page 55: 12 x1 t05 01 inverse functions (2013)](https://reader034.fdocuments.us/reader034/viewer/2022052622/5597ddd21a28ab58388b45fc/html5/thumbnails/55.jpg)
2xyiii y
x
2xy 21
xy Domain: all real x
Range:
21
21 :
xy
yxf
Domain:
Range:
0y
NO INVERSERestricted Domain:
Range: 0y
0x
0x0y
Book 2Exercise 1A; 2, 4bdf, 7, 9, 13, 14, 16, 19