Inverse Relations
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
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
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
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
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
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
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
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
Inverse Functions
Inverse FunctionsIf an inverse relation of a function, is a function, then it is called an inverse function.
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
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
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
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
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
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
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
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
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
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
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
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
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. 2
2xf xx
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
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
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
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
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
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
(ii) Draw the inverse relation
y
x
(ii) Draw the inverse relation
y
x
(ii) Draw the inverse relation
y
x
(ii) Draw the inverse relation
y
x
Exercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj