12x1 05 01 inverse functions (2011)
TRANSCRIPT
Inverse Functions
Inverse FunctionsIf y = f(x) is a function, then for each x in the domain, there is a maximum of one y value.
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)
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.
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
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.
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
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
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
Testing For Inverse Functions
Testing For Inverse Functions
(1) Use a horizontal line test
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi y
x
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi y
x
Only has an inverse relation
Testing For Inverse Functions
(1) Use a horizontal line test
e.g. 2xyi 3xyii y
x
y
x
Only has an inverse relation
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
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
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
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
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
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
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then;
xxff 1
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
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
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
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
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
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
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
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
Restricting The Domain
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.
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.
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
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
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
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
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
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
xeyii y
x
xey
1
xeyii y
xey
Domain: all real x
Range: y > 0 1
xeyii y
x
xey
Domain: all real x
Range: y > 0
xyexf y
log:1
1
xeyii y
x
xey
Domain: all real x
Range: y > 0
xyexf y
log:1
Domain: x > 0
Range: all real y
1
xeyii y
x
xey
Domain: all real x
Range: y > 0
xyexf y
log:1
Domain: x > 0
Range: all real y
1
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
2xyiii y
x
2xy
2xyiii y
x
2xy Domain: all real x
Range: 0y
2xyiii y
x
2xy Domain: all real x
Range: 0y
NO INVERSE
2xyiii y
x
2xy Domain: all real x
Range: 0y
NO INVERSERestricted Domain: 0x
2xyiii y
x
2xy Domain: all real x
Range: 0y
NO INVERSERestricted Domain:
Range: 0y
0x
2xyiii y
x
2xy Domain: all real x
Range:
21
21 :
xy
yxf
0y
NO INVERSERestricted Domain:
Range: 0y
0x
2xyiii y
x
2xy Domain: all real x
Range:
21
21 :
xy
yxf
Domain:
Range:
0y
NO INVERSERestricted Domain:
Range: 0y
0x
0x0y
2xyiii y
x
2xy Domain: all real x
Range:
21
21 :
xy
yxf
Domain:
Range:
0y
NO INVERSERestricted Domain:
Range: 0y
0x
0x0y
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
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